More homework, harder tests, and tough assignments, tutoring is all about an individualized plan that builds the academic skills, good habits and positive attitudes needed to succeed in high school and beyond.
This course extends students' experience with functions. Students will investigate the properties of polynomial, rational, logarithmic, and trigonometric functions; develop techniques for combining functions; broaden their understanding of rates of change; and develop facility in applying these concepts and skills. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended both for students taking the Calculus and Vectors course as a prerequisite for a university program and for those wishing to consolidate their understanding of mathematics before proceeding to any one of a variety of university programs. Prerequisite: Grade 11 Functions and Relations MCR3U
All functions must 'pass' the vertical line test with one value of y for each value of x.
Solution
Functions cannot have two values of y for one value of x. A vertical line drawn anywhere through a function will never cross a function twice.
Graphs of Functions: Even and Odd Functions
Determine whether the function is even, odd, or neither by comparing ƒ(x), ƒ(x), and ƒ(x).
Solution
Even Odd
Show algebraically whether the function is even:
Solution
All functions are even when
ƒ(x) = ƒ(x)
Substitute and see if above ^ is true...
Comparing the factored forms, the two functions are not equivalent. Since ƒ(x) ≠ ƒ(x), then the function is NOT even.
Show algebraically whether the function is odd:
Solution
All functions are odd when
ƒ(x) = ƒ(x)
Substitute and see if above ^ is true...
Since ƒ(x) ≠ ƒ(x), then the function is NOT odd.
End Behaviour
The end behaviour is what happens to y when x approaches positive infinity (+∞) or approaches negative infinity (∞).
The end behaviour is the same for the two functions:
Solutionƒ(x) = 5x^{9} + 3x^{4} + 2x^{3} ƒ(x) = 5x^{9}
The end behaviour depends only on the one term with the highest degree.
As x approaches positive infinity, y approaches negative infinity.
As x approaches negative infinity, y approaches positive infinity.
Determine the end behaviour of the function: ƒ(x) = 5x^{9} + 3x^{4} + 2x^{3} Solution
To determine the behaviour as x → ∞, try substituting a large/negative xvalue (like 100) and see if y → +∞ or y → ∞.
To determine the behaviour as x → +∞, try substituting a large/positive xvalue (like 100) and see if y → +∞ or y → ∞.
Determine which function has the end behaviour:
Solution As x → +∞, y → ∞ As x → ∞, y → ∞
To determine end behaviour try substituting large positive, and large negative values of x into the functions to see if y approaches +∞ or ∞.
Write each of the following as the other form... in either set notation or interval notation. Remember that interval notation uses parentheses ( ) for less than (<) or greater than (>), and square brackets [ ] is used for greater than or equal to [≥], and less than or equal to [≤].
±∞ ... uses ... ( )
>, < ... uses ... ( )
≥, ≤ ... uses ... [ ]
You will need some of the symbols below, provided for your copypaste convenience.
∞ ≤ ≥ < >
Write {XER  x ≥ 4} in interval notation.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Write {YER  23 < y ≤ 67} in interval notation.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Write ( 0, 4 ] in set notation in the form {XER  ______ }.
Solution
The inverse of which of the following is not a function?
Solution
The inverse of a quadratic function is not a function because it does not pass the vertical line test (i.e. it has two different values of y for the same value of x).
Inverse Functions
Determine the equation of the inverse of the quadratic function:
Solution y = x^{2}  6x  24
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Complete the Square
Converting to vertex form by completing the square would yield which of the following?
Solution Video y = ax^{2} + bx + c
Based on this work, determine the equation for the axis of symmetry.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
The equation of the axis of symmetry is the xvalue of the vertex...
Interestingly (as an aside) this is the equation of the max or min value...
Transformations with Function Notation
Which function is vertically compressed by a factor of ½, and translated 2 units right?
Solution
For the function:
y = a ƒ (k (x  d) ) + c
When a > 0, the function is vertically compressed by a factor of 'a'.
When d > 0, the function is translated 'd' units right.
From a quick observation or sketch, you will see that the piecewise function is not continuous, at x = 1.
State the domain and range, using proper set notation with curly braces { }, and pipe , like for example {XER  x > 5}.
Solution
Hint Clear Info
Domain:
Range:
Incorrect Attempts:
CHECK
Hint Unavailable
Domain: {XER}
Range: {YER  y < 5}
For the range, think about (or sketch) the greatest value of 'y' for each function in the piecewise notation. For ƒ(x) = 4x + 1, {XER  x < 1} you should see that the maximum 'y' value is 'less than' 5.
And of course for the function, ƒ(x) = 3, the maximum value of 'y' is 3 everywhere.
So putting this together the range of 'y' is everything less than 5, so it is {YER  y < 5}...
Polynomial Functions
Specific Topic
General Topic
School
Date
Solving a Volume Word Problem
Factoring Polynomials
North Toronto
Sep 2013
Intro to Polynomial Functions
The domain of a polynomial is always xεR, while the range has various bounds.
Solution
The domain can have various restrictions, or inequalities on it, and is not always just xεR.
Intro to Polynomial Functions
Which of the following is considered a polynomial function?
Solution
A polynomial cannot have:
• Negative exponents like 2x^{3}, or .
• Noninteger degrees like .
• Coefficients that aren't real numbers like
Real and Unreal (Imaginary, Complex Number) Roots of Polynomial Functions
The degree of a polynomial equals the number of real plus unreal roots. This rule does not work well for linear polynomials. Anyways, determine the number of roots in each...
For quartic (U or Wshaped) polynomials. (Some categories will apply to more than one graph)
Solution
4 real/equal roots
4 real roots
4 unreal roots
4 real roots (2 equal sets)
Drag each, to reveal correct answer.
For more quartic (U or Wshaped) polynomials. (Some categories will apply to more than one graph)
Solution
2 real, and 2 unreal roots
2 real/equal, and 2 unreal roots
2 equal/real roots, and 2 other real roots
3 equal/real roots, and 1 real root
Drag each, to reveal correct answer.
Finite Differences of a Polynomial
Use finite differences to determine the nature of the function.
Solution
x
y
3
88
2
29
1
6
0
1
1
4
2
27
3
86
The same number for 3^{rd} differences is a cubic function, like x^{3}...
Polynomial Functions and Finite Difference
Given the following formula for finite differences of a polynomial, where 'n' is the degree of the function,
Finite Difference = (Leading Coefficient) × (n!)
The polynomial, will have fifth differences the same.
Solution Finite Difference = (Leading Coefficient) × (n!)
The 5^{th} differences would take a long time to calculate, but it is done with the yvalues of consecutive xvalues...
The n^{th} differences are the same for the n^{th} degree. Where n = 5
If the 3rd finite differences of a different function are constant, with a difference of 12, then determine the value of the leading coefficient.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Remember
Finite Difference = (Leading Coefficient) × (n!)
If the 3rd finite differences of a different function are constant then the degree of the function is 3, so n = 3, and the given finite difference = 12...
Polynomial Function Properties
Polynomial functions never have vertical or horizontal asymptotes.
Solution
Rational functions have vertical and horizontal asymptotes,
Since rational functions are not polynomial functions, then polynomial functions never have vertical or horizontal asymptotes.
Symmetry of Polynomial Functions
Which of the following functions is neither even nor odd (referring to the degree)?
Solution
The rule for determining even and odd functions:
Even: ƒ(x) = ƒ(x) Odd: ƒ(x) = ƒ(x)
The following functions will do this because they only one even degree, which will cancel the negative sign, ƒ(x) = ƒ(x):
∴ a) and c) are even.
Check c)
∴ is odd.
Check d)
The different magnitudes 4, and 2 would not work with the other ƒ(x) = ƒ(x) either,
∴ is neither.
Given the information in the table below, determine the nature of the function.
Solution
x
ƒ(x)
4
664
3
33
2
23
4
74
5
43
A function is even when: ƒ(x) = ƒ(x).
A function is odd when: ƒ(x) = ƒ(x).
Use the points (4, 664) and (4, 74)
ƒ(+4) = 644, ƒ(4) = 74
∴ Not even, and not odd.
Determine the properties of the polynomial function...
Solution
If the function has symmetry on the vertical (y) axis, then it has an even degree. (But not all evendegree functions have vertical symmetry).
If the function has rotational symmetry about the origin or some other point, then it has an odd degree. (But not all odddegree functions have rotational symmetry).
Most evendegree functions (as compared to odddegree functions) have end behaviours on the same side of the xaxis.
Odd functions are either quadrant 2 & 4, or 1 & 3
Quadrant 2 & 4 is negative leading coefficient.
Quadrant 1 & 3 is positive leading coefficient.
Check the three correct properties of the polynomial function...
Solution
Even degree (because has line symmetry)
Negative leading coefficient (because even in quadrant 3 & 4)
Line symmetry (mirror image across the yaxis)
Check the two correct properties of the polynomial function, and state the factors, in your notes.
Solution
Odd degree, minimum degree of 5 (one more than the number of turning points)
Negative leading coefficient (because odd in quadrant 2 & 4)
Factors: (x + 3)(x + 1)(x)^{2}(x  4)
In the factors, it's (x)^{2} because the curve 'bounces' at this point, rather than goes straight through.
Characteristics of Functions
Which of the following functions has no zeros, one turning point, and an even degree?
Solution
Quadratics with x^{2} have only 1 turning point (because the maximum number of turning points in a function equals the degree  1).
This leaves only B) and D) as possible options.
You could try to factor but when you see neither quadratic is factorable, you could use the quadratic formula to determine the zeros. Furthermore, a function has no zeros/real roots/xintercepts when the value of the discriminant (b^{2}  4ac) is negative.
Try D) a=2, b=4, c=10
Try B) a=1, b=5, c=12
∴ Since the discriminant is negative in B), it will not have any real roots.
Polynomial Function Properties
Which of the following polynomials might have 5 turning points, without verifying by graphing?
Solution
The maximum number of turning points in a function equals,
= degree  1
A function with 5 turning points will have a degree of 6.
This leaves only:
Properties of Polynomial Functions
Determine which has an even leading coefficient and an odd degree.
Solution
A coefficient is a number infront of a variable. The degree of a function is the highest exponent on a variable.
Which of the following will have an odd leading coefficient and an odd degree? Solution
To determine the degree of an equation in factored form, add the exponents of all the common variables that are factored. The leading coefficient is determined by multiplying all the coefficients of the variables in factored form.
Zeros and Turning Points
How many turning points and xintercepts could the following function have?
Solution
Rule 1:
A polynomial function with degree n, has a maximum of n  1 turning points.
Rule 2:
The maximum number of xintercepts (real roots) is equal to the degree, n of the function.
Which of the following polynomial functions could not have 3 turning points?
Solution
A polynomial function with degree n, has a maximum of n  1 turning points.
Functions with degree of 4 or more could have 3 turning points.
The function would have 2 turning points at the most (and a minimum of 0 turning points).
Which of the following polynomial functions has an even number of turning points?
Solution
Polynomial functions with even degrees tend to have an odd number of turning points.
Polynomial functions with odd degrees tend to have an even number of turning points.
Degree and End Behaviour
Which of the following is the correct end behaviour of the function
Solution
This is just one end behaviour, of the righthand side. The end behaviour for the lefthand side is as x → ∞, y → +∞.
Which statement is incorrect regarding end behaviours of polynomial functions?
Solution
If the degree is even and the leading coefficient is negative, the ends are in the third and fourth quadrants.
The even degree means the ends are on the same side of the xaxis
A negative leading coefficient means the function is reflected vertically
Determine the end behaviours of the function below as x → +∞ and as x → ∞.
Solution
The degree of the function is odd, 5. Therefore the function has end behaviours on opposite sides of the xaxis.
Since the degree is odd and the leading coefficient is negative, the ends are in the second and fourth quadrants.
The function will pass through the xaxis at x = 0 and 4, and will bounce off the xaxis at x = 2.
You can get the xintercepts from the funtion.
At the xaxis, functions 'bounce' when the degree is even, and go through when the degree is odd.
To sketch the function the xintercepts, yintercept and which other aspect is required?
Solution
With the given information in the question, you are not sure how the function behaves above and below each of the xintercepts (1, 0) and (2, 0). You know the function will cross the xaxis at the point (1, 0) because the binomial has an odd exponent. And you know the function will bounce at (2, 0) because the binomial has an even exponent.
Having either an end behaviour or a point beside an xintercept will tell you the behaviour of the function at the xintercept, in other words the direction from which the function will cross or bounce at each xintercept.
The function has which of the following characteristics?
Solution
The xintercept occurs where the function equals zero:
From x^{3}... x = 0
From (x  3)^{2}... x = +3
From (x + 5)... x = 5
The yintercept occurs where x = 0...
Sketching Polynomial Functions
In your notebook, sketch a graph of the function
Solution
This gives all the information needed to make a proper sketch. (Exact turning points not required.)
Transformations of Polynomial Functions
State the transformations on the following polynomial.
Solution
Hint Clear Info
· Vertical compression by a factor of
· Reflection across the axis
· Horizontal compression by a factor of
· Translation unit(s) right
· Translation unit(s) up
Incorrect Attempts:
CHECK
Hint Unavailable
Factor out the 'k' value first.
Compare to:
The transformations:
Vertical compression by a factor of
Reflection accross the xaxis.
Horizontal compression by a factor of
Translation 1 unit right
Translation 10 units up
How can we verify if x = 2 is an xintercept of the polynomial:
Solution
The factor theorem says: (x  n) is a factor of ƒ(x) when ƒ(n) = 0.
For the following function (x + 2) is a factor.
Solution
(x + 2) is not a factor when there is a remainder (10). The factor theorem says: If (x  n) is a factor, then ƒ(n) = 0.
The function has xintercepts at x = 0, 5.
Solution
12 = x^{2}(x  5)
Nope! Expand and then use the factor theorem
Try different substitutions, until eventually f(2) = 0
Therefore, (x  2) is a factor.. using polynomial long division (not shown) you should get...
So x = 2... and using the quadratic equation (not shown) for x^{2}  3x  6...
Determine ƒ(5), ƒ(3) for ƒ(x) = 2x^{3}  3x^{2}  32x  15, and factor the equation...
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Using the factor theorem, if (x  n) is a factor, then ƒ(n) = 0.
Therefore (x  5) is one factor of ƒ(x)
Therefore (x + 3) is one factor of ƒ(x)
The divisor (4x  7) is a factor of when:
Solution
The factor theorem says: If (x  n) is a factor, then ƒ(n) = 0.
Determine the values of 'a' and 'c' given that (x  3) and (x + 2) are factors of ƒ(x) = ax^{3}  x^{2}  cx  12 .
Solution
Hint Clear Info
a = c =
Incorrect Attempts:
CHECK
Hint Unavailable
The factor theorem says: If (x  n) is a factor, then ƒ(n) = 0...
Solve the system of 2 equations with substitution or elimination... Sub ① in ②...
Factoring Polynomials: The Remainder Theorem
The remainder theorem equation, in corresponding form is:
ƒ(x) = d(x)·q(x) + r(x) dividend = (divisor)(quotient) + remainder
And in quotient form is:
If the function ƒ(x) has a factor of (x  n), then the remainder can be calculated with ƒ(n)...
Solution
Determine the remainder when is divided by (x  2).
Solution
remainder =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Substitute the root 2, in ƒ(2) to determine 'R'
(This question could also be solved with factor theorem, which is very similar to the remainder theorem)
A math student has used long division to determine the quotient and remainder of a polynomial. If the remainder is 0, the divisor is (x + 3) and the quotient is , then what dividend was used in the long division?
Solution
Note that when putting the dividend into long division, placeholders with a coefficient of 0 are required. should not be used. Instead use .
Determine the quotient in standard form, if the divisor is (x + 1), the remainder is 4, and the dividend is 3x^{3}  2x + 5.
Solution
⁰
¹
²
³
⁴
⁵
⁶
⁷
⁸
⁹
⁻
⁺
⁽
⁾
₀
₁
₂
₃
₄
₅
₆
₇
₈
₉
₋
₊
₍
₎
Incorrect Attempts:
CHECK
Hint Unavailable
When is divided by (x + 1) the remainder is 0. Determine the value of k.
Solution
k =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Given for (x + 1) the remainder is 0... n = 1...
(This question could also be solved with factor theorem, which is very similar to the remainder theorem)
If (x + 6) is a factor of , find the value of k.
Solution
k =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Given (x + 6) is a factor, the remainder equals zero when n = 6...
(This question could also be solved with factor theorem, which is very similar to the remainder theorem)
When is divided by (x  1) and (x + 5) it has the same remainder. Determine the value of n.
Solution
n =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
The Remainder Theorem Part II
Determine the remainder, without the quotient.
Solution
Dividend: ƒ(x) = 3x^{3}  2x^{2} + 10x  5
Divisor: (x + 4)(x + 3)
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
The remainder is always linear regardless of the degree of the dividend and divisor... It is either a constant, or has a degree of 1. Unless indicated otherwise, assume the remainder has a degree of 1, and represent as nx + m
From the root (x + 4), determine and substitute ƒ(4) = 269...
From the root (x + 3), determine and substitute ƒ(3) = 134...
Now solve the system of equations (using elimination here)...
Determine 'm'..
Therefore the remainder is: 135x + 271
Given the conditions below, determine the remainder, without the quotient or dividend, when the same divisor is divided into 3·ƒ(x)  2.
Solution
Dividend: ƒ(x)
Divisor: (x  1)
Remainder: 2
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
The question indicates that the remainder is a constant, so use one variable, 'R' to represent the remainder as a constant.
Substitute...
Now use this value for 3·ƒ(x)  2...
Factoring Polynomial Functions
The function is equal to which of the following?
Solution
Synthetic Division
Synthetic division works for all divisors.
Solution
Synthetic division only works for divisors that have one possible solution for 'x', for example:
(x + 1), (x  3), (2x + 3), ...
Not,
(x + 4)(x  4), (x  1)(x + 3), 2x^{2}  3x + 4
Solve for the remainder using synthetic division, given the divisor: (x + 2) Solution
ƒ(x) = 2x^{3} + 5x^{2}  x  6
A function passes through the point (1, 30) and has the xintercepts below. Determine the equation of the function.
Solution
First substitute the xintercepts, like...
Then substitute the point (1, 30) to determine the value of 'a'...
∴ the equation is:
Polynomial Word Problems
A rectangular prism is nested inside of an outer rectangular prism. The inner volume is
The volume of the space between the inner and outer prisms is
If the outer volume is 240 cm^{3}, determine the dimensions of the outer rectangular prism, in a commaseparated list.
Solution
dimensions:
Hint Clear Info
Incorrect Attempts:
CHECK
cm
Hint Unavailable
First determine an expression for the volume of the outer rectangular prism, in terms of x. Add the spacevolume to the innervolume,
Then determine one factor...
x = 3... Therefore the dimensions are:
(x + 5) = 8
(x + 3) = 6
(2x  1) = 5
Solving Polynomial Inequalities
Multiplying or dividing the variable by a negative requires the inequality sign to be reversed.
Solution
2x ≥ 4
True,
Solving Polynomial Inequalities
Solve the following inequalities with interval notation. Remember for interval notation:
The answer will contain the range where ƒ(x) ≤ 0.
The function has roots at x = 4, 5.
The yvalue, ƒ(x) is negative in the interval: {xεR  4 ≤ x ≤ 5}
Simplify the function:
The answer will contain the range where ƒ(x) ≤ 0.
The function has roots at x = 1, 1.
The yvalue, ƒ(x) is negative in the interval: {xεR  1 ≤ x ≤ 1}
The domain corresponds to the regions where y is positive (because of ≥ 0).
The end behaviours are in quadrant 1 and 3.
The function is tangent to the xaxis at x = 4.
Factor fully.
The roots occur at 3, 1, 1, 3.
Determine where the function is positive (Given from the interval " > 0").
The function has a degree of 4 (even) and a positive leading coefficient, therefore it has end behaviours in quadrant 1 and 2.
The function is positive on the interval: {xεR  x < 3, 1 < x < 1, x > 3}
Solve (and state your answers in increasing order).
Solution
Hint Clear Info
x_{1} = x_{2} =
Incorrect Attempts:
CHECK
Hint Unavailable
Cross multiply first to 'get rid of the fractions'..
Rational Equation Word Problems  Working Together
Solve the rational equation word problems, and remember the general form equation for problems on 'working together'.
Marcello takes 9 more minutes than Koji to make 10 pizzas. Working together, they can make 10 pizzas in 20 minutes. How long does it take Koji to make 10 pizzas when he works alone?
Solution
t =
Hint Clear Info
Incorrect Attempts:
CHECK
minutes
Hint Unavailable
Let 'x' be the time for Koji to make 10 pizzas.
Let 'x + 9' be the time for Marcello to make 10 pizzas.
Koji takes 36 minutes...
Sandi and John are computer programmers with different experience levels. It takes John 10 hours less than double the time it takes Sandi to write 2000 lines of code. Working together, it takes them 12 hours. How long does it take Sandi working alone?
Solution
t =
Hint Clear Info
Incorrect Attempts:
CHECK
hours
Hint Unavailable
Let 'x' be the time for Sandi to write 2000 lines of code.
Let '2x  10' be the time for John to write 2000 lines of code.
Takes Sandi 20 hours.. [Can't take Sandi 3 hours because then for John, 2(3)  10 would be negative time]
Recognizing Rational Functions
Which of the following is a rational function?
Solution
Rational functions must have polynomial functions in the numerator (dividend) and denominator (divisor). In other words, the quotient must be written with polynomials.
Polynomials cannot have square roots on the independent variable (but a rational function can have square root coefficients).
Rational Functions: Factoring, Xintercept(s), and Yintercept
Determine the 'x' and 'y' intercepts in coordinate form (x, y). Round your answers to one decimal place and list in increasing order, where applicable.
1) xintercept(s): solve for the numerator 2) yintercept: set x = 0 and solve for 'y'
Simplify & factor fully,
Using the quadratic equation to determine the xintercepts...
xintercepts: (1.8, 0), (3.8, 0)
Solve for yintercept with ƒ(0), if any...
yintercept: (0, 1.2)
Rational Expressions: Restrictions
Simplify and state the restrictions. (Enter your answers in increasing order, and where applicable leave as fully reduced fraction, instead of repeating decimals)
Enter in increasing order:
(Don't forget to state a restriction on anything that was ever a denominator, so even though the 7x1 gets flipped to the top, it still has to be included in the restriction list!)
Horizontal and Vertical Asymptotes: Degree of Numerator < Degree of Denominator
Determine the horizontal and vertical asymptotes of the following functions. (Order from low to high, where applicable)
The horizontal asymptote (HA) is always y = 0 + 'd' when the degree of the numerator is less than the degree of the denominator. (where 'd' is the vertical translation, not shown here)
Horizontal and Vertical Asymptotes: Degree of Numerator = Degree of Denominator
Determine the horizontal and vertical asymptotes of the following functions.
The horizontal asymptote (HA) is always when the degree of the numerator is equal to the degree of the denominator (where 'd' is the vertical translation, not shown here)
Vertical asymptote(s) @ x = ━━ , ━━
Horizontal asymptote(s) @ y =
Incorrect Attempts:
CHECK
Hint Unavailable
Horizontal and Vertical Asymptotes: Degree of Numerator > Degree of Denominator... (numerator = denominator + 1)
Determine the oblique (slant) and vertical asymptotes of the following functions.
There is no horizontal asymptote (HA) when the degree of the numerator is one more than the degree of the denominator. The slant/oblique asymptote is the quotient of
For the following function, which of the following statements is true?
Solution
Simplify,
A hole occurs when the same binomial is in the numerator and denominator, in this case: (3x + 2).
Though the (3x + 2) cancels in the denominator, there is still a restriction at x = 2/3.
You might think that there is a vertical asymptote at ⅔, or that a slant (oblique) asymptote exists at y = 2x  2, but holes win every time !
The graph just looks like a linear function of y = 2x  2, with a hole at x = .
Positive and Negative Intervals
Given the graph of the function
Determine the positive and negative intervals for the diagram of the function above.
Solution
Positive and negative intervals occur are based on where the function exists above or below the xaxis for positive or negative values of y.
Determine the intervals of increase and decrease for the diagram of the function above.
Solution
Think of the slope of the tangent at any one point. The intervals of increase have a tangent with a positive slope, and the intervals of decrease have a tangent with a negative slope.
Graphs of Rational Functions: Domain, Range, IncreaseDecrease Intervals, and PositiveNegative Intervals
Determine the positive and negative intervals of the following function, using set notation.
Solution
Hint Clear Info
Positive: { XER  }
Negative: { XER  }
Incorrect Attempts:
CHECK
Hint Unavailable
Reciprocal Coordinate
A point P(4, 21) exists for the following function . Determine the corresponding coordinate of the reciprocal of this function.
Solution
The xvalue is the same, while the reciprocal of the yvalue is
Reciprocal of Absolute Value Function
The reciprocal of the absolute value function y = ¼x  2 has vertical asymptote(s) at
Solution
Graphing the Reciprocals of Linear Functions
For the reciprocal of the function , determine:
The vertical and horizontal asymptotes, and the domain and range.
Solution Video
To find positive/negative intervals substitute xvalues on either sides of the vertical asymptotes into ƒ(x), and check if function is positive or negative...
Since the vertical asymptote is x = 5/4, check the intervals: x<5/4, and x>5/4...
For x < 5/4...
The function is negative: {XER  x < 5/4}
For x > 5/4...
The function is positive: {XER  x > 5/4}
Make a sketch to visualize the increasing/decreasing intervals, using...
HA: y = 0
VA: x = 4,
As x → ∞, y → 0
As x → +∞, y → 0
Determine the axis of symmetry of the local maximum, either from the vertex of the original function or from the midpoint of the vertical asymptotes.
After a sketch, you can see...
Increasing: {XER / x < 4, 4 < x < }, or (∞, 4) ∪ (4, )
Decreasing: {XER / < x < , x > }, or (, ) ∪ (, ∞)
Graphing the Reciprocals of Quadratic Functions
For the reciprocal of the function , determine:
The vertical and horizontal asymptotes, and the domain and range.
Solution
The xintercepts, positive/negative intervals, and increasing/decreasing intervals.
Solution
Determine the end behaviour of the function.
Solution
From the previous sketch, see that: x → ∞, y → 0, x → +∞, y → 0
Reciprocals of Quadratic Functions: The Relationship of Roots and Vertical Asymptotes
For the reciprocal of the following functions, determine if any vertical asymptotes exist.
Hole at x = 3, horizontal asymptote at y = 0, and vertical asymptote at x = 5.
Solution
A vertical asymptote exists at x = 10, and a xintercept of x = 5, given the following function:
Solution
Describing Graphs of Rational Functions
Where applicable, determine the increasing, decreasing, positive, and negative intervals.
Solution
Common factor,
Determine the features:
VA: x =
HA: y = 2 (when degree of numerator & denominator are equal: use ratio of leading coefficients)
yintercept: y =
xintercept: x =
Make a table to determine the positive/negative intervals (note: 0 is not positive or negative)
(∞, )
(, )
(, ∞)
(2  3x)
+
+

(2x  1)

+
+
Results:

+

Positive: {XER / < x < }, or (, ]
Negative: {XER / > x > }, or (∞, ) ∪ [, ∞)
Make a quick sketch in your notes (not shown) to determine the intervals of increase and decrease
Increase: none
Decrease: {XER / x ≠}, or (∞, ) ∪ (, ∞)
Reciprocal Function Word Problem: Substitution
The concentration of an antinausea remedy in the body can be determined with the following function where ƒ(t) is concentration in mg/L, and t is time in hours. Determine the time from initial dosage, when the concentration in the body is 1.40 gm/L.
Solution
t =
Hint Clear Info
Incorrect Attempts:
CHECK
hours
Hint Unavailable
Solve using the quadratic formula, where a = 2.8, b = 23.6, and c = 16.8
Move the terms to the same side, simplify, and factor,
To find where the function ƒ(x) is negative (< 0)... Don't worry about the binomial that results in the imaginary number (2x^{2} + 1). Set up an interval chart with ranges between any of the roots and vertical asymptotes (if any),
Move the terms to the same side, simplify, and factor,
To find where the function ƒ(x) is negative (< 0)... Set up an interval chart with the ranges between any of the roots (x = 2) and vertical asymptotes (x = 3, +3)
Determine the value of the reciprocal trig ratio,
Solution Video
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Explain the difference between a reciprocal and an inverse of the following...
Solution
The reciprocal divides '1' by the 'y'values of the function. For instance [not shown], asymptotes are formed where the function had y = 0...
The inverse represents the angle,
Although this seems correct, the answer cannot have a square root in the denominator! To derationalize the denominator, multiply the top and bottom by the square root:
Use the unit circle...
Principle angle = 240˚...
In quadrant 3, cos is negative...
Related acute angle = 60˚...
Using special triangles...
cos of 60˚...
= 1/2
Characteristics of Graphs of Trig Parent Functions
Which of the following is not a similarity between the functions ƒ(x) = sinθ and ƒ(x) = cosθ?
Solution
Furthermore the following features are also similar: amplitude, range.
The only differences are the xintercepts and yintercepts.
Characteristics of Graphs of Trig Parent Functions
How many times do the functions ƒ(x) = sinθ and ƒ(x) = cosθcross the xaxis in the domain 360˚ ≤ θ ≤ 360˚, respectively?
Solution
Hint Clear Info
&
Incorrect Attempts:
CHECK
Hint Unavailable
How many times do the functions ƒ(x) = sinθ and ƒ(x) = cosθcross 'eachother' in the domain 360˚ ≤ θ ≤ 360˚?
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Characteristics of Graphs of Trig Parent Functions
The period of ƒ(x) = sinθ over the domain 360˚ ≤ θ ≤ 360˚ is not 720˚, rather it is 360˚ or 2π radians.
Solution
The period is not determined by the domain. Rather, the period is always the measurement of the xaxis over just one cycle.
Characteristics of Graphs of Trig Parent Functions
The domain range, and period of the three main trig functions are given below. Fill in the blanks.
Solution
sin(θ)
cos(θ)
tan(θ)
Domain
x∈ℝ
x∈ℝ
x ≠ + n·π, n∈I, n = {...1, 0, 1, ...}
Range
y∈ℝ, 1 ≤ y ≤ 1
__________
y∈ℝ
Period
2π
2π
π
Hint Clear Info
y∈ℝ, ≤ y ≤
Incorrect Attempts:
CHECK
Hint Unavailable
Make a quick sketch of cos(θ)... see that the maximum is +1 and the minimum is 1, so the range is: y∈ℝ, 1 ≤ y ≤ 1
State the xintercepts in periodic notation in terms of pi, for the function f(x) = sin(θ).
Solution
Remember, or make a quick sketch, and see that some of the xintercepts are: 0, π, 2π, so the xintercepts are..
π·n, n∈I
(n∈I means 'n' is all real integers)
The graph of tan(θ) has vertical asymptotes... therefore the amplitude is infinity, or undefined...
Reciprocal Trig Functions
Which of the following is not a reciprocal of the function, ƒ(x) = cosθ?
Solution
The inverse of cos, cos^{1}(θ) is not the same as reciprocal! Just think of reciprocal as flipping, with 1 over it...
Order of Transformations of Trig Functions
Determine the recommended order to apply the transformations according to order of operations. Sort by dragging, starting with the first operation at the top.
Solution Videoy = a · sin (k(x  c)) + d
The amplitude is 2, and the axis of symmetry is shifted down to y = 2.
So the maximum is y = 0, and the minimum is y = 4.
Transformations of Trig Functions
Determine which of the following is equivalent to the function, .
Solution
Solve by graphing the transformations to check overlapping functions, or solve by plugging in points in radians and comparing the outputs, which is faster. Plug in and if the functions are equivalent, the outputs will be the same.
Therefore the functions and are equivalent.
Transformations of Trig Functions
Which of the following set of transformations would transform the graph of ƒ(x) = cos(x) into the function below.
Solutiong(x) = 2cos(3x  ) + 1
Vertical stretch by a factor of 2, horizontally compressed , translated right, and translated 1 unit up.
Transformations of Trig Functions
Given the function,
Determine all the transformations from the parent cosine function.
Solution
Hint Clear Info
· Reflection in the axis
· Vertical compression by a factor of ──
· And a horizontal compression by a factor of ──
· Translated units
· Translated units
Incorrect Attempts:
CHECK
Hint Unavailable
· Reflection in the xaxis
· Vertical compression by a factor of
· Horizontal compression by a factor of
· Translated units right
· Translated 2 units down
Determine the value of ƒ(x) when x = , and reduce fully.
Solution
Hint Clear Info
ƒ(x) = ───
Incorrect Attempts:
CHECK
Hint Unavailable
Period of Trig Functions
Given the period of a trig function equals 20, determine the exact value of k, in radians. Solutiony = a · sin [k(x  c)] + d
ƒ(x) =
Hint Clear Info
Incorrect Attempts:
CHECK
radians
Hint Unavailable
For the period, (T = 20)
Transformations of Trig Functions
The period of the function, ƒ(x) = 5cos(4x + 2), in radians is:
Solution
period =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
ƒ(x) = a•sin[k(x  d)] + c
k = 4
Solving for the Independent Variable in the argument of a Trig Function
Given the function below...
Solve for xalgebraically, in radians, rounded to the nearest hundredth decimal place.
Solution
x =
Hint Clear Info
Incorrect Attempts:
CHECK
rad
Hint Unavailable
Determine the next value of 'x', rounded to the nearest hundredth decimal place, where the function equals 5, over the interval: 0 ≤ x ≤ 0.5π
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Using the cast system, sin is positive (1/2), at π/6 and 5π/6... So
And,
Features of Trig Functions
Determine the features of the function below. Reduce your answers fully.
A large ferris wheel at a circus has a diameter of 50m with a platform to mount it at the lowest point 2m above ground level. One full ride around the wheel takes 4 minutes. Find the following
The equation of the trig function, H(t) with height starting at the bottom of the ferris wheel at t = 0.
Solution
H(t) =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
If the ferris wheel starts at the bottom, then height = 2m when t = 0.
Word Problems with Trig Functions: Clock
The hour hand on a clock rotates at 1 revolution per 12 hours. The clock has a diameter of 4cm and the point of interest to model is located at the tip of the hour hand, at the perimeter of the clock. Model the following using a sine function with start time at 12:00 noon.
You need to determine the kvalue for the equation:
Word Problems with Trig Functions: Tides
Narragansett Bay has a maximum tide height 5m higher than the lowest point of 1m. A low tide point occurs at 2:00PM. If the time between adjacent high and low tides is 6 hours, 15 minutes, determine the following...
"If the time between adjacent high and low tides is 6 hours, 15 minutes"... this is half a cycle. Therefore the period is double this... 12 hours and 30 minutes... 12.5 hours...
The horizontal translation, d (using a positive cos function, starting at 12:00 noon)
Solution
d =
Hint Clear Info
Incorrect Attempts:
CHECK
ƒ(x) = a·cos[k(x  d)] + c
If we are to use a cos function instead of a sine function, then we will use the maximum point of the tide as our starting point. We will set time as zero at 12:00 noon. So, with a low at 2:00PM there is a high tide 6 hours, 15 minutes later at 8:15PM. This is 8 hours, 15 minutes or 8.25 hours after noon. So, the horizontal translation will be 8.25 right
Word Problems with Trig Functions: Seasonal Temperatures
A function modelling the average temperature T, in degrees Celsius, in a certain ountry over the course of one year is given below where t is the time in days. Determine the following
The maximum temperature, and the month this occurs in. [3]
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Maximum Temperature = 16 + 13 = 29˚ Celsius
Solve to find this occurs in April.
The 190th day is ~ 6.3 months ~ June...
What is the average temperature for the entire year?
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
˚C
Think of the axis of symmetry
The average is the axis of symmetry, which is 13 (degrees Celsius).
Predict the temperature on May 21^{st}, the 140^{th} day of the year.
Solution
T =
Hint Clear Info
Incorrect Attempts:
CHECK
˚C
Hint Unavailable
Substitute t = 140 into the equation...
This evaluates to 23.4 degrees Celsius.
Exact Value of Compound Angle Formula Using Principle Angles of Special Related Acute Angles
Evaluate each of the following with an exact value, using the compound angle formulas. These questions use the principle angles of related acute angles that are special triangle angles.
Use the compound angle formula: cos(x˚ + y˚) = (cos x˚)(cos y˚)  (sin x˚)(sin y˚)
You need to find combinations of angles that have 'Related Special Acute Angles' so you can lookup the exact values.
Note this could also work or be done with cos(60˚ + 225˚)... you would get the same answer.
Compound Angle Formula (aka Addition Subtraction Formula)
The double angle formula is derived partly using the compound angle formula, see:
The double angle formula just uses double or twice the angle in the trig, for example:
sin(2x), or cos(2x), or tan(2x).
If cos(x) = , then cos(2x) = over the interval 0 ≤ x ≤ π.
Solution
False.
Given tan(x) = , determine sin(2x) and cos(2x) over the interval 0 ≤ x ≤ π. Reduce fully.
Solution
Hint Clear Info
sin(2x) = ━━cos(2x) = ━━
Incorrect Attempts:
CHECK
Hint Unavailable
First determine the other two ratios using reflex angle triangles on the CAST system, from 0 to π... This would be in quadrant 1 only. After making a sketch, you can determine that,
Next sub this into the double angle formulas to solve,
and,
furthermore,
Evaluate over the interval 0 ≤ x ≤ π, giving exact answers. State your answers in increasing order.
sin(4x) = cos(2x)
Use the double angle formula on sin(4x),
The double angle, '2x' changes the interval from 0 ≤ x ≤ π, to 0 ≤ 2x ≤ 2π... Using a sketch of the cos function, and the CAST system...
There are 4 answers, in increasing order:
Double Angle Formula Identities
Determine an equivalent expression to the following.
Determine sin(60˚), in terms of 'm' or 'n', given the following.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Using the CAST system, see that cos(300˚) = cos(60˚) because they share the same (acute) reflex angle, and are both positive (in quadrant 1 & 3).
Using the CAST system, see that cos(300˚) = cos(60˚) because they share the same (acute) reflex angle, and are both positive (in quadrant 1 & 3). So,
Enter answer as "1  m^2"
Solve the trig equation on the interval: 0 ≤ x ≤ .
Solution
x =
Hint Clear Info
Incorrect Attempts:
CHECK
radians
Hint Unavailable
The interval is in radians, so put your calculator in radians.
Solve the trig equation on the interval: 0 ≤ θ ≤ 360˚. Order from lowest to highest.
Solution
Hint Clear Info
θ_{1} = θ_{2} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
Use CAST: sinθ is positive in Q1 & Q2 (sketch not shown), so using corelated angles...
Solving Trig Equations
Solve for the exact value in degrees, on the interval: 0˚ ≤ x ≤ 90˚
Solution
x =
Hint Clear Info
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
Solving Trig Equations
Solve using degrees, on the interval 0˚ ≤ x ≤ 360˚. Order your answers from lowest to highest.
Solution
Hint Clear Info
x_{1} = x_{2} = x_{3} = x_{4} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
In the function that equals zero, either of the following is zero:
(2sin x  1) = 0 OR,
(cos^{2} x) = 0
sin(x) is positive, using CAST and special triangles we get x = 30˚, or 150˚
Use your understanding of a cos(x) graph. cos(x) = 0 at the xintercepts x = 90˚, or 270˚
Solving Trig Equations
Solve using degrees on the interval: 0˚ ≤ x ≤ 360˚. Order your answers from lowest to highest.
Solution
Hint Clear Info
x_{1} = x_{2} = x_{3} = x_{4} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
Here we must factor by grouping (common factor the first two terms, then common factor the last two terms). This is the trickiest part of the question for most.
cos is negative in quadrant II and quadrant III x = 150˚, 210˚sin is positive in quadrant I and quadrant II x = 60˚, 120˚
Solving Trig Equations
Solve using degrees on the interval: 0˚ ≤ x ≤ 360˚. Order your answers from lowest to highest.
Solution
Hint Clear Info
x_{1} = x_{2} = x_{3} = x_{4} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
It may be tricky to see that this is a difference of squares!
tan is positive in quadrant I and quadrant III x = 30˚, 210˚
tan is negative in quadrant II and quadrant IV x = 150˚, 330˚
Solving for the Trig Equation
Determine the quadratic equation, in standard form, that would evaluate to x = 240˚ and 300˚ on the interval 0˚ ≤ x ≤ 360˚.
Solution
240˚ and 300˚ both have related acute angles of 60˚
Located in quadrant III and IV, where sin is negative...
Put into factored form of quadratic equation and foil to get standard form:
Solving Trig Equations with Various Arguments
Solve for the interval: 0˚ ≤ x ≤ 90˚. Order your answers from low to high.
Solution
Hint Clear Info
x_{1} = x_{2} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
(Common Mistake) Remember, because the angle is '2x', the interval changes by the same factor, 2,
0˚ ≤ x ≤ 90˚ > 0˚ ≤ 2x ≤ 180˚
Using CAST: see that sin(x) is positive in Q1 & Q2.
And the principle angles in Q1 & Q2 are,
Solve for the interval: 0˚ ≤ x ≤ 1,080˚. Order your answers from low to high.
Solution
Hint Clear Info
x_{1} = x_{2} = x_{3} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
(Common Mistake) Remember, because the angle is , the interval changes by the same factor, divide by 2,
0˚ ≤ x ≤ 1080˚ >
Using CAST: see that tan(x) is positive in Q1 & Q3. (sketch not shown) See that this is special triangles with 'x' = 1, 'y' = , and 'r' = 2, with the reflex angle of 60˚
Now find all other (principle) angles, including coterminal angles up to 540˚
Solving Trig Equations
Solve using degrees, on the interval 0˚ ≤ x ≤ 360˚, and determine the total number of answers.
Solution
It may be tricky to see that this is a difference of squares!
Tan is negative in quadrant II and IV, and positive in quadrant I and III.
Polynomial Trig Equations
Solve over the interval 0˚ ≤ θ ≤ 360˚. State your answers in increasing order.
Solution
Hint Clear Info
θ_{1} = θ_{2} =
Incorrect Attempts:
CHECK
degrees
Hint Unavailable
Simplify,
Let x = sinθ
Using polynomial long division (not shown)...
Sub back, x = sinθ
Now do each 'CAST' separately to determine the principle angles over the given interval (sketches not shown)...
Arrange the solution steps shown here into the correct order (by dragging them).
Solution
The square root of 7 can be written as an exponent:
The exponent in a log comes down and multiplies infront of the log:
If the base of a log is the same as the number in brackets, this always equals 1:
These are all the same, it doesn't matter what base is chosen.
According to the change of base formula below, 2log_{5}(12) is equal to which of the following?
Evaluate log_{4}(23) to the nearest tenth decimal place.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Basically, it's impossible to type a nonbase10 log into the calculator. So you need to change the base to base 10 first, and then type into the calculator.
Use change of base formula and plug into calculator.
A colony of bacteria doubles every n number days according to the function below. If the colony starts with 88 bacteria and turns into 22,528 in 20 days, determine the doubling time of the colony.
Solution
n =
Hint Clear Info
Incorrect Attempts:
CHECK
days
Hint Unavailable
Word Problems: Decibel Sound Intensity
Sound intensity is measured in units of decibels (dB), where 'I' is sound intensity and 'I_{0}' is a reference intensity of the threshold of our hearing ability.
A car crash is typically 145 dB, while the airbag is typically 165 dB. How many times more intense is the sound of the airbag than the sound of the crash itself?
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
times
Hint Unavailable
First rearrange the equation for intensity 'I' using log laws
Then determine the ratio, using exponent laws
The airbag is 100 times more intense sounding than the crash.
Natural Log and Natural Exponential (Base 'e')
'e' is a real number, approximately equal to 2.71828, that is a base of natural logarithms.
Solve. Round your answer to the nearest hundredth decimal place.
Solution
x =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
So, either,
Natural Log Word Problem
The metabolism of a certain drug by the average male is given by k = 0.1234 hr^{1}. If the initial concentration is 9 mg/L and the final concentration is 0.5 mg/L, determine the time it takes.
Solution ln[A]_{f} = kt + ln[A]_{i}
t =
Hint Clear Info
Incorrect Attempts:
CHECK
hours
Hint Unavailable
Word Problems with Logs
$10,000 is invested at 3% annually over a certain number of investment periods, n. If the final value is $11,592.74 determine the number of periods n.
Solution
Which of the following is the inverse of the function ?
Solution
Switch 'x' and 'y' and then isolate for 'y'.
Using the change of base formula,
Graphing Logarithmic Functions
Prove that the following functions are inverse using the coordinate (1, 3).
Solution
Inverse functions switch 'x' and 'y' in the coordinates (a, b) ↔ (b, a).
Therefore the above functions are inversely related.
Graphing Logarithmic Functions
The xintercept of the function y = log(x) occurs at:
Solution
Note the log law:
The xintercept always occurs when y = 0
The log of 1 is always zero.
In other words, the xintercept occurs at (1, 0).
Graphing Logarithmic Functions
The vertical asymptote of y = log(x) occurs at x = 0.
Solution
True, see:
Graphing Logarithmic Functions
Describe the transformations on the function:
Graphing Logarithmic Functions
Transform y = log(x) to produce a function with the following transformations
Solution
Vertical stretch by a factor of 3
Reflection in the yaxis
Horizontal stretch by a factor of 2
Horizontal translation 4 units right
Vertical translation 1 unit up
Graphing Logarithmic Functions
Which property do the following functions all have in common?
Solution
They all have a vertical asymptote at x = 0...
Graphing Logarithmic Functions
Which one of the following functions does not have the same domain as the others?
Solution
All have the domain {xεR, x > 0} except the function that has the domain {xεR, x > 1} because it is translated 1 unit left, which changes the location of the vertical asymptote.
Graphing Logarithmic Functions
Determine the exact value of the xintercept of the function .
Solution
x =
Hint Clear Info
━━
Incorrect Attempts:
CHECK
Hint Unavailable
The xintercept occurs at y = 0.
Working with Log Equations
Solve, rounding your answer to the nearest hundredth.
Solution
x =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
For an in initial investment of a certain amount (A_{0}) at a certain rate (R) over a certain amount of time (t), different compounding periods (n) will have a very small difference in the final amount (A) of an investment.
The following are approximately equivalent:
"If you put $1 into a bank account paying 100% interest compounded continually, at the end of the year, you’ll have exactly e dollars."
A $100.00 investment is turned into $112.75 over a 3 year period. If the interest rate is compounded monthly, calculate the interest rate.
Solution
rate =
Hint Clear Info
Incorrect Attempts:
CHECK
%
Hint Unavailable
Cannot solve for rate with the equation:
This equation must be used when solving for rate:
(Notice that the compounding period is not used)
Rates of Change of Functions
Rates of Change
Which of the following scenarios would have a constant, positive average rate of change?
Solution
A positive rate of change has a positive slope. A constant average rate of change has the same magnitude of slope for all intervals of 'x'.
Instantaneous and Average Rates of Change
Given the function ƒ(x) = x^{4}  x^{3}, find:
the average rate of change on the interval 3 ≤ x ≤ 1
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
the instantaneous rate of change at x = 5
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Instantaneous Rate of Change at the Vertex of a Quadratic Function (Without Difference Quotient)
For the function ƒ(x) = 3(x  4)^{2} + 1
Estimate the instantaneous rate of change at the vertex, without using difference quotient.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint: Slope at the vertex point.
Determine the average rate of change from x = 2 to x = 8.
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Exact Instantaneous Rate of Change using the Difference Quotient
Given the function ƒ(x) = 2x  4x^{2}, find the slope of the tangent using the difference quotient, at x = 2.
Solution
m =
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Exact Instantaneous Rate of Change using the Difference Quotient
A baseball is launched into the air and its height can be modelled by the function h(t) = 5t^{2} + 15t + 1, where h(t) is height in metres, and t is time in seconds. Determine the exact instantaneous rate of change in the height of the baseball at 2s.
Solution
s =
Hint Clear Info
Incorrect Attempts:
CHECK
m/s
Hint Unavailable
Givens:
t = 2.0 s
h = 0.01
For a small 'h' = 0.01 approaching zero,
Now substitute into the original equation, and solve,
Verify if a Maximum or Minimum Exists at a Given Point Using Estimation of Instantaneous Rate of Change
Given the function ƒ(x) = x^{2}  4x + 5, verify and state if a maximum or minimum occurs at the point (2, 1). [1]
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Verify Whether a Point is either a Maximum or a Minimum
Given that a point of inflection occurs at x = 2 for the following function, ƒ(x) = 2x^{3} + 7x^{2} + 4x, verify and state whether the point is a maximum or a minimum. [1]
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Rates of Change with Trig Functions
Average Rate of Change for a Trig Function
For the function ƒ(x) = 8sin(x), calculate the exact value of the average rate of change on the interval [0, 90]
Solution
Hint Clear Info
━━
Incorrect Attempts:
CHECK
Hint Unavailable
Estimating Instantaneous Rate of Change for a Trig Function
For the function ƒ(x) = 6sin(30x) + 10, calculate the instantaneous rate of change at x = 2
Solution
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Combinations & Compositions of Functions
Combination & Composition Operations and Notation
Given the functions:
f(x) = 3x + 2
g(x) = 5x^{2}  2x
The functions can be combined in which way?
Solution
Functions can be combined with any of the operations listed in the question. You will see examples of this later on.
ƒ ∘ g is equivalent to which two of the following operations or notations?
Solution
The following are equivalent:
ƒ ∘ g = (ƒ ∘ g)(x) = f(g(x))
The following are equivalent:
(f × g)(x) = f(x) × g(x)
Combination and Composition with Tables of Values
Determine each of the following, using the table given below.
Determine the set (of coordinates) contained within the composition ƒ[g(x)].
Solution
x
g(x)
2
1
1
5
0
4
1
3
2
0
3
4
x
ƒ(x)
2
2
1
3
0
6
1
17
2
23
3
12
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
The set (of coordinates) contains the points with the 'x' value from g(x) and the 'y' value from f(x), in the form: (x_{g}, y_{f}). This comes from the points that share the same 'y' value from g(x) substituted as 'x' for f(x)...
Since,
f[g(2)] = 17
And,
f[g(2)] = 6
Then the set is:
{(2, 17), (2, 6)}
Determine whether the following is true or false.
Solution (ƒ — g)(x) = ƒ(x) — g(x)
This is combination notation.
If n(x) = 12 and m(x) = 6, then (n + m)(x) = 16.
Solution
Determine the following, given ƒ(x) = 4x + 2 and g(x) = 2sinx.
Solution(ƒ  g)(45)
Hint Clear Info
Incorrect Attempts:
CHECK
Hint Unavailable
Combine the functions first, (ƒ  g)(x) = ƒ(x)  g(x), then substitute the independent value,
Combinations: Product, Trig
Given (ƒ × g)(x) = 8x sinx + 4sinx, what are two possible values of ƒ(x) and g(x)?
Solution
Domain of Combinations
Which statement is true about the domain of all combinations of functions (added, subtracted, multiplied, or divided).
Solution
You know that domain is the range (or region) of independent values that can exist for a function. The domain of a combination of functions will be determined by the shared region of 'x' that exists.
If the domain of function 1 is {XER  1 ≤ x} and the domain of function 2 is {XER  x ≤ 10}, then the domain of the combined function is {XER  1 ≤ x ≤ 10}.
Solution
The combined domain is the region of x that is shared by both functions.
State the domain for the (ƒ × g)(x) of the following functions, using proper set notation.
Solutionƒ(x) = sin(x) g(x) = log(x)
Hint Clear Info
Domain:
Incorrect Attempts:
CHECK
Hint Unavailable
The domain of ƒ(x) = sin(x + 2) is {XER}
The domain of g(x) = log(x) is {XER  x > 0}
So the combined domain is {XER  x > 0}.
Domain of Compositions
Which of the following compositions have the same domain for (ƒ ◦ g)(x) and (g ◦ f)(x)?
Solution
None of the functions have the same domain. Some may think that and have the same domain but, the domain of is {XER: x ≥ 0} and the domain of is {XER}. [Both have the same range, {YER: y ≥ 0}]
As an aside: and should have the same domain, {XER}
Solving Combination of Functions (Intersection)
The point of intersection of two functions can be solved with guess and check.
Solution
1^{st} set the equations equal.
2^{nd} guess a value of x that makes it close to LS = RS
3^{rd} Sub x into your equation to solve for y and the point.
Combinations
Determine which of the following functions could create the look:
Solution
It looks like a trig function that is increasing exponentially.
Combination with Division (Quotient)
The (ƒ ÷ g)(x) will have restrictions based on the value of x in g(x).
Solution
When g(x) is in the denominator, the combined function will have restrictions on all values of 'x' for which g(x) equals zero.
State the domain on the combination of (f ÷ g)(x) where,
Solutionƒ(x) = 2x + 3 g(x) = log(x)
log(x) ≠ 0, so x ≠ 1
log(x) > 0, so x > 0
(interestingly in this case, there is a restriction in a restriction.)
The domain is: {XER  0 < x < 1, x > 1}
{XER  x ≠ 0, 1} is not considered correct because 'x' cannot be negative.
Determine two possible functions, m(x) and n(x) that compose to form p(x) = 2^{3x + 4} Solution
(Answers may vary) One possible answer is:
Sub n(x) into m(x) to get p(x).
Combination of Functions with Unknowns
The two points (1, 24), (3, 0) lie on the combined function of, (n × m)(x). Determine the values, 'a' and 'b'.
Solution
n(x) = 2x^{2} + ax  3
m(x) = bx^{2}  5x + 3
Hint Clear Info
a = b =
Incorrect Attempts:
CHECK
Hint Unavailable
The domain of the combined function is shared (the same) for both n(x) and m(x).
This allows us to use the 'x' value from the combined function, '1' and '3', and substitute into both n(x) and m(x).
We do this to get the 'y' values for n(x) and m(x).
For the combined function, (n × m)(x), the 'y' values multiply so that: y_{n} × y_{m} = y_{combined}.
We show this below.
Substitute the 'x = 1' into the equations for n(x),
And the 'x = 3', into m(x),
The combination of functions, (n × m)(x), is the product of the 'y' values of n(x) and m(x), y_{n} × y_{m} = y_{combined}:
Substitute one equation for another. You can isolate for 'a' in one equation
And substitute it into the other,
Plug in each 'b' value into one of the previous equations to solve for 'a'
Therefore, b = 4, and a = 5
[b ≠ 2 since 'a' is undefined for that value].