This course enables students to broaden their understanding of relationships and extend their problem-solving and algebraic skills through investigation, the effective use of technology, and abstract reasoning. Students will explore quadratic relations and their applications; solve and apply linear systems; verify properties of geometric figures using analytic geometry; and investigate the trigonometry of right and acute triangles. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Grade 9 Principles of Math MPM1D
TABLE OF CONTENTS
Linear Systems
Solution to a System
The solution to a system is the point of intersection. Solution
'Solve the system' means find the point of intersection (POI). The solution to the system is the POI.
Solutions
Match the systems to their number of solutions.
Solution
0 (no) solution
1 solution
2 solutions
Infinite solutions
A system has 0 (no) solution when the lines are parallel (same slope).
Infinite solutions occur when the lines overlap (collinear).
1 solution is 1 point of intersection.
Simplify equation ①
Simplify equation ②
Convert equation ① × 3 and equation ② × 5 for elimination...
Using elimination solve for y...
Solve for x...
Linear System Word Problems
Solve.
[Geometry] In triangle ABC, angle B is 20 less than three times angle A, and angle C is 20 more than two times angle A. Find the angles A, B, and C.
Solution
Hint Clear Info
A = B =C =
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degrees
Hint Unavailable
Let angle A be 'A'
Let angle B be 3A - 20
Let angle C be 2A + 20
Angle B,
Angle C,
[Proportions] A 4.0 L solution contains 20% alcohol, and is mixed with an 8.0 L solution that is 50% alcohol. What percent of the resulting mixture is alcohol?
Solution
Hint Clear Info
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%
Hint Unavailable
Percent alcohol is the total amount of alcohol,
... divided by the total volume,
[Money] For a concert, Sarah decides to order T-shirts for all of the participants. It costs $4 per shirt for the medium size, and $5 per shirt for the large size. Sarah orders a total of 70 T-shirts and spends $320. How many are medium shirts?
Solution
Hint Clear Info
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Let 'x' be the number of medium shirts
Let 'y' be the number of large shirts
The amount of money made from each shirt type is (# sold)(price)
Equation ① comes from the total number of shirts sold...
Equation ② comes from the total money made...
Solve by substituting ① into ②...
Therefore there are 30 medium sized shirts.
[Money Alternate] Jeremy and Sarah have a combined income of $80,000. One quarter of Jeremy's income is the same as one-sixth of Sarah’s income. How much does each person earn?
Solution
Hint Clear Info
Jeremy: $ Sarah: $
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Hint Unavailable
Let 'x' be Jeremy's income
Let 'y' be Sarah's income
Equation ①
Equation ②
Solve substituting ① into ②...
Solve for 'y'...
[Distance, Speed, Time] A train leaves Town A and heads for Town B at an average speed of 45 km/hr. A second train leaves Town B at the same time and heads for Town A at an average speed of 30 km/hr. If the route is 450 km long, then how many hours will it take for the two trains to meet?
Solution Video
Hint Clear Info
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hours
Hint Unavailable
Write the givens (d, s, t) in a table to organize...
Train A
Train B
Distance
x
450km - x
Speed
45 km/h
30 km/h
Time
t
t
Notice the time for Train A equals time for Train B...
The distance Train A travels is 270km... Now calculate the time for train A...
Train A (and Train B) take 6 hours to meet.
[Alternate Distance, Speed, Time] A boat took 5.0 hours to travel 60 km up a river against the current. The return trip took 3.0 hours. Find the speed of the boat in still water and the speed of the current.
Solution Video
Hint Clear Info
Boat speed: Current speed:
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km/hr
Hint Unavailable
Make a table to organize the distance, speed, and time for the upstream and downstream trips...
Let 'b' be the speed of the boat, and 'c' the speed of the current.
Upstream
Downstream
Distance (km)
60 km
60 km
Speed (km/h)
b - c
b + c
Time (h)
5 h
3 h
Make an equation for d = s × t for the upstream trip...
Make an equation for d = s × t for the downstream trip...
Multiply ① × 3 and ② × 5 for elimination...
Eliminate the c's to solve for 'b', by adding ② & ①...
Then, solve for c...
[Proportions] If a nickel costs the US mint 9.4 cents to make and a dime costs 4.6 cents, determine the unit cost of nickel and copper, given the information below.
Solution
A nickel weighs 5 grams and is 25% nickel and 75% copper
A dime weighs 2.268 grams and is 8.33% nickel and 91.67% copper
Hint Clear Info
Unit cost nickel: ¢Unit cost copper: ¢
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Hint Unavailable
Let 'x' be the cost of nickel (per unit mass)
Let 'y' be the cost of copper (per unit mass)
The mass amount of a specific metal is the mass of the whole coin times the ratio proportion.
The total cost of the metal is the mass amount times the cost (per unit mass)
Equation ① for the nickel coin,
Equation ② for the dime coin,
Solve using substitution... Rearrange ①...
Substitute into equation ②...
Solve for 'y'
Analytic Geometry
Line Segments
Explain the difference between a right bisector and an altitude. [3]
Solution
Hint Clear Info
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Hint Unavailable
Equation of a line given two points
Determine the equation of the line connecting the two points, D(-2, 1) and G(2, 10). (Reduce improper fractions fully).
Solution
y =
Hint Clear Info
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Hint Unavailable
Equation of the right bisector of a line given two points
Determine the equation of the right bisector of a line connecting the two points T(-8, 6) and U(2, 10). (Reduce improper fractions fully).
Solution
y =
Hint Clear Info
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Hint Unavailable
Find equation of a right bisector and verify if a point lies on the line.
Does the point P(-3, -2) lie on the right bisector of the line segment with endpoints Q(-2, 5) and R(4, 1)? Justify your answer, by showing your work in your notes.
Solution
Hint Clear Info
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Hint Unavailable
Pre - Length of a line formula
The distance between two points can be calculated as the hypotenuse using the Pythagorean theorem. Given the two points A(1, -5) and B(3, n)...
Use the Pythagorean theorem to determine the distance between the two points, in terms of n.
Solution
Use a sketch to visualize/determine the change in x and the change in y...
∆ x = 3 - 1 = 2
∆ y = n - (-5) = n + 5
Calculate hypotenuse length (distance), c...
Now, calculate the length of the line between these two points using the distance formula below, in terms of n.
Solution
Make A(1, -5) point 1, and B(3, n) point 2.
x_{1} = 1
y_{1} = -5
x_{2} = 3
y_{2} = n
Takeaway: See that the distance (length) of a line equation is based on the same idea as the Pythagorean theorem. It is important to understand this before you jump into the distance (length) of a line equation, so you know where it comes from.
This is true because the square of the differences for example, (5 - x)^{2} or (x - 5)^{2}, becomes the same positive magnitude...
Which of the following operations is/are correct?
Solution
II is wrong because when there are more than one term underneath the square root, then the root and squares no longer cancel. The root only cancels the squares when there is one square term underneath (example I).
Length of a line formula
On a street map, the coordinates of the two fire stations in a town are A(10, 63) and B(87, 30). A neighbor reports smoke coming from the kitchen of a house at C(41, 18). Which fire station is closer to this house?
Solution
Hint Clear Info
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Hint Unavailable
Two firestations are located at A(-10, -3) and B(6, -9). Station A is old with an outdate line of 'Firemobile' firetrucks that can only safely go a top speed of 40 km/h in city traffic. Station B is a completely new station with new 'FS1' firetrucks with GPS, AWD, and ABS technology and that can safely go a top speed of 65 km/h in city traffic. There is a fire at F(-4, 5) where the coordinates are in kilometers. Determine the outcome assuming everything runs smoothly.
Solution
Determine the distance of the fire from station A...
Determine the distance of the fire from station B...
Time for Firemobile from station A...
Time for FS1 from station B...
Therefore Firemobile trucks from station A arrive first.
Perpendicular lines, solving systems, and length of a line segment.
A line connects points X(1, 2) and Y(5, 7). A third point Z(-2, 8) is located nearby.
Determine the equation of the line connecting XY.
Solution
y =
Hint Clear Info
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Hint Unavailable
Solve the system of the two perpendicular lines, XY and the perpendicular line from Z, ie. determine the point of intersection (POI). Simplify fully.
Solution
Hint Clear Info
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Hint Unavailable
Calculate the shortest distance from point Z to the line XY.
Solution
d =
Hint Clear Info
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Hint Unavailable
Equal Segments (Distances)
Determine the coordinates of the two points that split the line connecting the points (10, 20) and (37, 8) into three equal segments.
Solution
Hint Clear Info
(19, ) & ( , )
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The total difference in 'x' is...
The 3 equal segments have distances in 'x'...
The total difference in 'y' is...
The 3 equal segments have distances in 'y'...
The two points are: (19, 16) & (28, 12).
∆ = +9: 10 ↔ 19 ↔ 28 ↔ 37
∆ = -4: 20 ↔ 16 ↔ 12 ↔ 8
Equation of a Circle at the Origin
Determine whether or not the point P(√35, 2) lies inside, outside, or on the perimeter of the circle defined by the function...
Solutionx^{2} + y^{2} - 50 = 0
Hint Clear Info
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Hint Unavailable
A circle with its center at the origin, passes through the point (7, 5). Determine the equation of the line that is tangent to this point. Reduce fully.
Solution
Hint Clear Info
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Hint Unavailable
The tangent is the negative reciprocal of the radius at the point (7, 5). The slope of the tangent is
The slope of the tangent, the negative reciprocal is,
Calculate the b-value, by plugging in the point...
So the equation is,
Circle Word Problems
Pollini's is a famous pizza restaurant located in a major city that delivers within a 12km radius from the restaurant location. A hungry group of business people wants to order from this restaurant and they are located 8km west and 6km south of the restaurant. Will the restaurant deliver to these people?
Solution
Hint Clear Info
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Hint Unavailable
Set the pizza place at the origin (0, 0) of a Cartesian plane, the 8km west and 6km south would be (-8, -6)... Determine the hypotenuse (or the radius of the circle)...
The business people are located 10 km away from the restaurant, therefore yes, the restaurant will deliver them.
Equation of a Circle, Shifted
Given the equation for a circle with a center at (h, k)...
Determine the equation of the circle with a center at (-4, -3) and a radius of .
Solution
⁰
¹
²
³
⁴
⁵
⁶
⁷
⁸
⁹
⁻
⁺
⁽
⁾
₀
₁
₂
₃
₄
₅
₆
₇
₈
₉
₋
₊
₍
₎
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Hint Unavailable
Given the equation of the circle, confirm that the points (-3, 11) and (3, 11) lie on endpoints of a chord of the circle.
Solution
Chords:
Endpoints of the chord are on the perimeter of the circle
The chord is less than the diameter (otherwise it is the diameter)
Confirm by substituting points into circle equation and checking if left side equals right side.
And again,
Therefore both points form a chord on the circle with center at (0, 4).
Calculate the area of the triangle formed between the x-intercept points and the maximum point on the circle... given the equation of the circle below.
Solution(x - 5)^{2} + (y - 1)^{2} = 65
A =
Hint Clear Info
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units^{2}
Hint Unavailable
See from the equation that the center of the circle is at: (5, 1)
Determine the maximum height at x = 5...
Determine the x-intercepts by setting y = 0...
Calculate the area... with the x-intercepts forming the base and the max height 9.06 as the height...
Determine the equation of the tangent to the point (1, 5) on the circle with the equation below. Reduce fully.
Solution
Hint Clear Info
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Hint Unavailable
Complete the square for x and y in the circle equation...
The slope of the tangent is the negative reciprocal of the slope between the center (-1, 2) and point (1, 5) ...
Calculate b using ⊥m and a point...
The equation of the tangent is
Applications of Circles
Built in the 1400's, the Duomo in Florence, Italy is still the largest masonry dome in the world. The sides of the arched Duomo lie on part of the perimeters of two circles shown below, where the intersection forms the cross section of the dome. Determine the height of the Duomo if the radius of each circle is 43 m and the base diameter of the Duomo is 45 m.
Solution
Creative Commons: Sailko, 2006
h =
Hint Clear Info
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m
Hint Unavailable
Put the x and y axis through the center of the base of the Duomo. This makes the point of intersection (POI) at x = 0 in the form...
(0, y)
According to the origin (0, 0), determine the center of a circle to make an equation. See that (45 - 43) gives us the horizontal distance between a center of the circle and the other circles nearest point on the perimeter. Using the circle on the right side, the center of the circle is located at (20.5, 0)
The circle on the right side has its center translated from the origin by (h, k), which corresponds to the points we determined (20.5, 0).. The equation of the circle is:
From where the origin was placed, you can see the POI is at x = 0, now solve for y using the equation of the circle...
The height of the Duomo is 38 m. (Only use the positive value for y).
Applications of Circles, Chords, Distance
A volcano in Iceland erupts with center C(10, 14), the ash cloud formed has radius 3 km and can be modeled with the circle equation below. A weather drone plane flies on route y = 0.8x + 9.
Determine if the weather drone will enter the ash cloud. Prove your reasoning by showing your work in your notes.
Solution
Hint Clear Info
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Hint Unavailable
Determine the point(s) of intersection (POI), if any...
Expand the circle equation and substitute the linear equation into it...
Sub the linear equation y = 0.8x + 9...
Use the quadratic equation, where a = 1.64, b = -28, c = 116...
Since there are 2 POIs then drone flies through the ash cloud...
The weather drone manufacturers built the drone with a warranty to withstand flying through a maximum of 3km of ash conditions. Determine if the drone should divert its course.
Solution
Hint Clear Info
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Hint Unavailable
In other words, find the coordinates of the POIs to calculate the distance.
First find the coordinates by substituting the 'x' values into the equation...
The points are (7.07, 14.66) and (10, 17)
Calculate the distance (the length of the chord)...
The drone would fly 3.75 km through the ash cloud, and therefore should divert its course to remain under warranty.
Geometric Properties
Line Segments and Centers
Right bisectors are used to determine the centroid of a triangle.
Solution
Right bisectors are used to determine the circumcenter of a triangle (and circle).
Altitudes are used to determine the orthocenter of a triangle
Midpoints through the opposite vertex are used to determine the centroid of a triangle
Orthocenters and circumcenters can be located outside of their triangles, while centroids cannot.
Solution
This is true in obtuse triangles. Orthocenters and circumcenters can be located outside of their triangles. Centroids are never located outside of their triangles.
Centers
State any one relationship between the three: circumcenter, centroid, and orthocenter.
Solution
Hint Clear Info
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Hint Unavailable
Any one of the following...
In equilateral triangles, the circumcenter, centroid, and orthocenter all overlap at one point.
The circumcenter, centroid, and orthocenter are always arranged forming a line.
The centroid is always located between the circumcenter and orthocenter.
The centroid-orthocenter distance is always double the centroid-circumcenter distance.
Equations of Lines
Determine the equation of a line that passes through point C and the midpoint of AB, given the points: A(1, 2) B(3, 4) and C(5, 6).
Solution
y =
Hint Clear Info
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Hint Unavailable
Find the midpoint of AB...
Find the slope between the midpoint (2, 3) and point C(5, 6)...
Determine 'b' using slope and any point on the line...
So the equation is...
Triangles and Distance
Given the following points, determine if triangle ABC is either scalene, equilateral, or isosceles.
Solution A(-5, 2) B(4, 2) C(2, -5)
Hint Clear Info
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Hint Unavailable
Determine the 3 side length distances...
3 equal side lengths = equilateral
2 equal side lengths = isosceles
no equal side lengths = scalene
AB...
AC...
BC...
Since none of the side lengths are equal, the triangle is scalene.
Right Bisectors
A right bisector is located at the midpoint on a line, and is perpendicular to the slope of the line. Given the following points, determine the equation of the right bisector of QR.
Solution P(3, -1) Q(5, 5) R(0, 3)
Hint Clear Info
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Hint Unavailable
A right bisector is at the midpoint, and is perpendicular to the side.
Determine the midpoint of QR...
Determine the ⊥ slope of the side QR...
Determine the equation using ⊥m and the midpoint on the line...
The equation of the right bisector of QR is...
Circumcenter of a Triangle
Given the following points, determine the circumcenter of the triangle. (Hint: a circumcenter is where all the right bisectors intersect.)
L(2, 2) M(1, 1) N(0, -3)
Determine the equation of the right bisector of LM.
Solution
y =
Hint Clear Info
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Hint Unavailable
With a quick sketch you will see that this triangle is obtuse rather than acute... An you will remember that with obtuse triangles, the circumcenter is located outside of it...
Circumcenters are right bisectors... Find the midpoint...
Find the ⊥m of LM...
Equation of right bisector of LM...
Determine the equation of the right bisector of MN.
Solution
y =
Hint Clear Info
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Hint Unavailable
With a quick sketch you will see that this triangle is obtuse rather than acute... An you will remember that with obtuse triangles, the circumcenter is located outside of it...
Circumcenters are right bisectors... Find the midpoint...
Find the ⊥m of MN...
Equation of right bisector of MN...
Determine the coordinate of the circumcenter. Reduce fully.
Solution
Hint Clear Info
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Hint Unavailable
The circumcenter is the point of intersection, POI... Find the point of intersection of the right bisectors of LM and MN...
Substitute ① in ②
Determine y...
Therefore the circumcenter is located at
Circumcenters in Circles
A large firestation is to be constructed to service three small municipalities nearby. The city councils require that each city center of each municipality is the exact same distance to the firestation. The city centers are located at...
A (5, 9)
B (-10, -9)
C (15, -10)
Determine the coordinate, location where the firestation should be built. Reduce fully.
Solution
Hint Clear Info
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Hint Unavailable
Place each point on the perimeter of the circle surrounding the firesation at the center. Determine the centroid with the intersection of any 2 medians (using midpoints) from the sides of the triangle...
Midpoint 1... AB:
Midpoint 2... AC:
Slope from Midpoint_{AB} to C
Slope from Midpoint_{AC} to B
Equation of 1... using m_{AB} & C
Equation of 2... using m_{AC} & C
POI of 1 and 2... (use substitution or elimination)
The firestation should be built at:
Determine the equation of the circle that can be drawn through the city centers around the firestation.
Determine the circumcenter using the midpoints, then get the radius using the distance from a vertex coordinate to the circumcenter coordinate.
Sub into the equation for h, k, and r...
Quadratic Relations
Parabolas: a Simple Sketch First
Sketch the parabola for the following...
Given the table of values for the equation,
y = x^{2} - 4x + 4
x
y
-2
16
-1
9
0
4
1
1
2
0
3
1
4
4
5
9
6
16
Given just the equation,
y = -3x^{2} + 2
Hint, make your own table of values...
x
y = -3(x)^{2} + 2
bbb
ccc
bbb
ccc
bbb
ccc
bbb
ccc
And then sketch...
Degree of Expressions with one Variable
Determine the degree of each of the following expressions.
(As you can see (4)^{6} is not incorrect, but can be simplified further when 4 is replaced with 2^{2}. You will see simplifying the base fully becomes more useful later).
Exponents: Combinations
Write with a single, positive exponent, showing your work without using a calculator.
Solution
Hint Clear Info
━━━
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Hint Unavailable
Number Sense: Exponent Rules with Variables: Power Rule with Coefficients
The following simplification of exponents is correct.
Solution
True. This is called the power rule.
Number Sense: Exponent Rules with Variables, Including Negative Exponents
Simplify, answers should have positive exponents only.
Remember back to the optimization you learned last year. The vertex is located at the maximum or minimum of the y-values. In this table, there is a maximum located at (4, 17).
Determine the equation of the axis of symmetry Solution
x =
Hint Clear Info
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The equation of the axis of symmetry is located at the x-value of the maximum point, which is:
The maximum height is the highest y-value, which is 17.
Don't forget to write the units:
Therefore the maximum height occurs at 17m.
Verify that the equation h = -x^{2} + 8x + 1 can be used to model the flight path of the ball. [1]
Solution
Hint Clear Info
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Hint Unavailable
You verify an equation by substituting 2 or more points into the equation and check that L.S. equals R.S.
Yes, the equation can be used to model the flight path of the ball.
Finite Differences (First and Second)
Use finite differences to determine if the following relations are linear, quadratic, or neither.
Calculate the first difference by subtracting each adjacent y-value.
4 - 6 = -2
6 - 8 = -2
8 - 10 = -2
10 - 12 = -2
All first differences are the same, therefore the relation is linear.
-19 - (-12) = -7
-12 - (-7) = -5
-7 - (-4) = -3
-4 - (-3) = -1
The first differences are not the same, therefore not linear. Find second differences:
-7 - (-5) = -2
-5 - (-3) = -2
-3 - (-1) = -2
Since the second differences are the same, then the relation is quadratic.
x
y
1
0
3
1
5
8
7
27
9
64
Hint Clear Info
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Hint Unavailable
0 - 1 = -1
1 - 8 = -7
8 - 27 = -19
27 - 64 = -37
Quadratic Functions
A basketball shot is taken from a horizontal distance of 5 m from the hoop. The height of the ball can be modeled by the equation below. Where h is height in meters, and t is time in seconds since the ball was released.
From what height was the ball first released?
Solution Video
h =
Hint Clear Info
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m
Hint Unavailable
The ball was first released when t = 0 s.
If the ball reached the hoop at 1.0 s, what was the height of the hoop?
Solution Video
h =
Hint Clear Info
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m
Hint Unavailable
Determine the height h(t) when t = 1 s.
What was the maximum height reached by the ball?
Solution Video
h =
Hint Clear Info
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m
Hint Unavailable
Use the formula for the axis of symmetry at the vertex...
a = -7.3, b = 8.25, c = 2.1
The y-value in the vertex (0.57, 4.43) gives us the maximum height. Therefore the maximum height is 4.43 m.
Given the ball is 4m above the ground at 0.32 seconds, determine the other time when it is at 4m.
Solution
t =
Hint Clear Info
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s
Hint Unavailable
Solve using symmetry. Given the point (0.32, 4) and the vertex (0.57, 3.05) you can find the next time.
This is symmetrical with the other side... add 0.25 to the 0.57
The other point is (0.82, 4)
Translations and Vertex Form
Given the general, vertex form of a quadratic.
Which of the following values determines a vertical compression by a certain factor?
Solution
Vertical compression when: 0 < |a| < 1.
Vertical stretch when: |a| > 1.
Which of the following values determines a reflection in the x-axis?
Solution
Reflection on x-axis when: a < 0... when 'a' is negative.
Parabola opens up when z > 0... when 'a' is positive.
Which of the following values determines the horizontal shift left or right?
Solution
Careful, this is the tricky one!
+h = shift right
-h = shift left
Transformations of Quadratic Functions
Match the type of transformations - shown with equations - to their respective graphs.
Solution
Vertical stretch or compression by a factor of 'a'...
Horizontal shift/translation by 'h' units left or right...
Vertical reflection...
Vertical shift/translation by 'k' units up or down...
Transformations of Quadratic Functions
State the transformations on the following quadratic functions compared to the parent function, .
Given the general form of a quadratic, determine the equation of a quadratic function with the transformations listed below....
Reflection in the x-axis, vertical stretch by a factor of 3, vertical translation 5 units up.
Solution
A parabola with vertex at (-1, 3), opening upward, and with a vertical compression by a factor of ⅛.
Solution
Hint Clear Info
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Hint Unavailable
Transformations of a Point Quadratic Functions
Transform the point (1, 2) according to the same transformations in the function below. The point (1, 2) transforms into which of the following?
Solution y = -2(x + 1)^{2} - 3
Hint Clear Info
( , )
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Hint Unavailable
y = -2(x + 1)^{2} - 3
Reflection in the x-axis. ← Affects y
Vertical stretch by a factor of 2. ← Affects y
Translation 1 unit left. ← Affects x
Translation 1 unit down. ← Affects y
Two Other Transformations of Quadratic Functions
Based on the value 'n' determine some of the horizontal transformations below.
The transformation to y = a[n(x - h)]^{2} + k with a horizontal reflection.
Solution
A horizontal reflection is due to ƒ(-x), or in an other form, y = a[-n(x - h)]^{2} + k.
For example consider the transformation of to .
The transformation to y = a[n(x - h)]^{2} + k with a horizontal stretch by a factor of 5. (careful here)
Solution
The vertical stretch or compression is by a factor of !
So for a horizontal stretch by a factor of 5, the 'n' value must be ⅕ ... because .
Sketching Graphs of Functions
Sketch the graphs of the following functions. Label the x-intercepts, the axis of symmetry, and the vertex for each.
Roots at x = 1, -5.
Axis of symmetry at x = -2
Vertex: (-2, -18)
Solving Quadratic Functions
A soccer ball is kicked upward on the field. The height, ‘h’, of the ball in meters is given by the function below, where ‘t’ is the time in seconds. Find the maximum height and the time the ball reaches its maximum height.
Solution
Hint Clear Info
Height = m
Time = s
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Hint Unavailable
The maximum height occurs at the y-value of the vertex. Solve by completing the square:
Vertex: (1, 10)
The maximum height is 10 m at a time of 1s.
Solving Quadratic Functions
Given the x-intercepts of a function are -1 and 3, and the y-intercept is 6, the equation of the parabola is
Solution
Put the x-intercepts into the general form of the quadratic in standard form...
The y-intercept (6) occurs where x = 0 ...
ƒ(0) = 6 ...
So putting this together we get,
Solving Using Symmetry in the Quadratic Relations Unit
A parabola has vertex (-3, 7), and one x-intercept is -11. Find the other x-intercept and the y-intercept (reduce fully).
Solution
Hint Clear Info
x-intercept:
( , ) y-intercept:━━━
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First find the other x-intercept with symmetry.
x = +5
The x-intercept is at (5, 0)
To find the y-intercept, determine the equation first... plug in the points (-3, 7) and (-11, 0)
Then set x = 0...
Solving Using Quadratic Relations, and Vertex Form
A quadratic function passes through the points (-1, 1), (0, -3), and (5, 1).
You want to solve the system of equations. First substitute the easiest point which has a zero in it, (0, -3) the y-intercept into standard form...
Substitute 'c' and the point (-1, 1) in...
Substitute 'c' and the point (5, 1) in...
Substitute b = a - 4 into the other equation to solve for 'a' and 'b'...
Solve for 'b'...
So the equation is,
First determine the Axis of Symmetry (AOS) of the quadratic. The AOS goes through the vertex and is the midpoint of two points on the same horizontal line: (-1, 1) and (5, 1).
Substitute AOS and the easiest point (0, -3) into the vertex form... This AOS represents 'h' in the equation...
Substitute AOS and any other point, (-1, 1)...
Solve with elimination...
Now find the last value, 'k by subbing in the last point (5, 1)...
Here is your equation in vertex form,
For factoring quadratics where there's nothing but 1 in-front of the x^{2}...
The numbers that multiply to 12:
1 × 12, 2 × 6, 3 × 4
Choose the pair that adds to 7... 3 × 4
= (x + 3)(x + 4)
As with the rest of the questions here, you can factor by grouping (common factor the first and last pairs). Basically apply what you already know (common factoring).
Two numbers that add to +4 and multiply to -60... -6 & +10...
It is good form to common factor first actually, although common factoring at the end is acceptable.
Best to common factor first. What two numbers add to +2 and multiply to -15... +5 & -3...
Faster Factoring Without Decomposition
Now that you understand more about factoring it's time to take the training wheels off - no more decomposition (aka splitting and grouping). Factor straight into the parenthesis like the example shown below.
What multiplies to +18 and adds to +11? = +9 and +2
Which of the following is a perfect square trinomial?
Solution
Perfect squares are in the form: a^{2} + 2ab + b^{2}, or a^{2} - 2ab + b^{2}
Algebraic Expressions and Factoring
The dimensions for two rectangles are given below. Determine an expression for the total area of the two rectangles combined, then combine like terms and factor this expression fully.
Solution
Rectangle A
Rectangle B
Length (cm)
(2x - 3)
4
Width (cm)
3x
(x - 1)
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Add the areas,
Then factor the expression,
Algebraic Expressions and Factoring
A triangular sandbox is surrounded by a rectangular field of grass. The dimensions of the sandbox and field are listed below, in terms of x.
Base (cm)
Height (cm)
Length (cm)
Width (cm)
Triangle
2x
x + 4
Rectangle
x + 1
4x - 4
Determine a standard form expression for the area of the grass surrounding the sandbox.
Solution
The volume of a rectangular prism is represented by the following trinomial. Determine a factored-form expression for the dimensions in terms of x with the length, width, and height.
Solution Volume = 2x^{3} + 8x^{2} + 6x
V =
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V = (l) × (w) × (h)
Algebraic Expressions and Factoring
Determine the most simplified expression for the total surface area of a cube with side lengths equal to (2x - 3).
Solution
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There are 6 equal faces on a cube. The area of each face is (2x - 3)(2x - 3) = (2x - 3)^{2} The total surface area = 6(area) = 6(2x - 3)^{2}
Quadratic Equations
Specific Topic
General Topic
School
Date
Solving for X by Completing the Square
Factoring Quadratics
Branksome
Sep 2013
Solving Factored Quadratics Using Zero Product Property
Determine the solutions, hence solve. Use the zero-product property or complete the square where most applicable. Order solutions from lowest to highest...
Since not factorable, solve by completing the square...
Solutions and Vertex
Given the quadratic: y = 4x^{2} - 36,
Determine the x-intercepts, hence factor.
Solution
Hint Clear Info
x_{1} = x_{2} =
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The x-intercepts are:
x = -3, +3
Determine the vertex, hence show in vertex form.
Solution
Hint Clear Info
( , )
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This form is basically already in vertex form,
The vertex is (0, -36).
Algebraic Equations and Factoring
A rectangle has the following dimensions: length = (2x - 5), and width = (4x + 2).
Write an expression for the area of the rectangle, in standard form.
Solution
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If the area of the rectangle equals 14cm^{2}, then find the dimensions of the sides in centimetres.
Solution
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Notice that this questions is not asking for a "maximum" or "minimum"...
So you are looking for the roots...
To find the dimensions you need to find x-intercepts not x-value of vertex:
Determine the Vertex: Complete the Square
Rewrite the equation in vertex form by completing the square.
Solution Video y = x^{2} + 4x + 6
Determine the Vertex: Complete the Square
Rewrite the equation in vertex form by completing the square.
The equation of the axis of symmetry is: x = - 3 Solution y = -5x^{2} - 30x + 12
x = -3
The equation of the axis of symmetry is: x = - 3
Vertex Form
The minimum occurs at...
Solution y = -3x^{2} + 6x - 20
y =
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Determine the minimum 'y' value (of the vertex)
One of the main ways to determine a maximum/minimum is to convert the quadratic equation from standard form to vertex form. The minimum is the y-value of the vertex.
Complete the square:
y = -3x^{2} + 6x - 20
y = -3(x^{2} - 2x) - 20
y = -3(x^{2} - 2x + 1 - 1) - 20
y = -3(x^{2} - 2x + 1) - 20 + (-3)(-1)
y = -3(x^{2} - 2x + 1) - 17
y = -3(x - 1)(x - 1) - 17
y = -3(x - 1)^{2} - 17
The maximum point is (1, 20) Solution y = 8x^{2} - 16x + 20
The function opens upwards because +8x^{2} is postive, therefore the vertex is a minimum point.
Convert Vertex Form to Standard Form
Convert to standard form: y = -3(x - 2)^{2} + 4
Solution
y =
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y = -3(x - 2)^{2} + 4
y = -3(x - 2)(x - 2) + 4
y = -3(x^{2} - 2x - 2x + 4) + 4
y = -3(x^{2} - 4x + 4) + 4
y = -3x^{2} + 12x + -12 + 4
y = -3x^{2} + 12x - 8
Maximum and Minimum Word Problems
The trajectory of a dud firecracker launched by astronauts on a planet with oxygen and high gravity is recorded with height in meters and time, t in seconds. What is the maximum height of the firecracker on the planet?
Solution
h(t) = -2t^{2} + 10t + 1
h =
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m
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The maximum height is the y-value of the vertex.
The maximum height is 13.5m
Revenue, Expense, and Profit
The revenue, expense, and profit functions are given below, where x is the number of items sold.
Make a profit equation, and factor fully.
Solution
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Determine the number of items sold at the break even point (when profit equals zero).
Solution
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items
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Profit equals zero so substitute '0' into the left hand side of your profit equation:
Solve for x when y = 0 by factoring, rather than completing the square to find vertex. x cannot be negative (because you can't have a negative number of things sold.)
∴ the break even point occurs when 5 items are sold.
Maximum and Minimum Word Problems
The cost of operating a machine, in dollars, is given by the formula below where t is time, in hours, that the machine operates.
C(t) = 2t^{2} - 16t + 682
What is the minimum cost of running the machine
Solution
$
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The minimum cost is the y-value of the vertex.
The minimum cost is $650
For how many hours must the machine run to reach this minimum cost?
Solution
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hours
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The hours is the x-value of the vertex.
C(t) = 2(x - 4)^{2} + 650 The machine runs for 4 hours.
Maximum and Minimum Word Problems
A hamburger store can sell 120 hamburgers per week at $4.00 per hamburger. For each $0.50 decrease in price, they can sell 20 more hamburgers.
Determine an algebraic expression in standard form, for the maximum revenue.
Solution
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Let 'x' represent number of times the price changes by $0.50
What price should the store charge to maximize revenue?
Solution
$
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To solve for price ($4 - $0.50x), you need to find the value of 'x' of the vertex.
You can find the x-value by completing the square: (note that you can also find the x-value of the vertex by symmetry, or the equation x = -b/2a).
Word Problem: Solving for Elements of Quadratic Equations
Given the following elements of a quadratic function, write the equation in factored form.
Solution
Axis of Symmetry:
= -4
x-intercept:
(-6, 0)
y-intercept at:
y = 12
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Givens:
Root 1: (-6, 0)
Point 2: (0, 12)
Using symmetry, Root 2: (-2, 0)
Word Problem: Solving for Elements of Quadratic Equations
Write an equation, in standard form, to represent a parabola with:
Solution
Word Problem: Solving for Elements of Quadratic Equations
Write an equation, in vertex form, to represent a parabola that passes through the origin and the points: and
Solution
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Substitute each of the three coordinates into equation (in vertex form) to make three equations in order to solve the system...
Equation ①: substituting ...
Equation ②: substituting ...
Equation ③: substituting ...
Substitute equation ① into ② and substitute equation ① into ③ to get two equations for h...
Now that you have 2 equations and 2 unknowns (h & a) you can use substitution to solve the system.
Set h = h ... hence solve for a...
Solve for h...
Solve for k...
Finally after all that algebra practice! Your equation is...
Quadratic Equations Theory
Explain what must be true for a parabola to have only one x-intercept.
Solution
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There will only be one solution for x, one point.
One root means the quadratic equation will turn out to be a perfect square when completing the square...
When setting ƒ(x) = 0, the discriminant will equal zero...
Deriving the Quadratic Formula
Derive the quadratic formula from the general quadratic equation in standard form. Show your work in your own notes.
Solutionax^{2} + bx + c = 0
Complete the square with a quadratic in standard form: ax^{2} + bx + c = 0...
To solve the trinomial, 2x^{2} - 7x = -4, which cannot be factored we must use the quadratic equation. The quadratic equation has been used correctly below.
Solution
Quadratic Formula
When using the quadratic formula, what indicates that there are no x-intercepts?
Solution
This is the discriminant (b^{2} - 4ac). Since it is underneath a square root, a negative value indicates 'no real roots' (no x-intercepts)
Mistakes to Avoid Solving Equations
In the statement below, 'x' equals which of the following values?
Solution
Vertex to Factored Form
The following equation in vertex form represents the size of the opening of the arch of the Basilica of Maxentius and Constantine in Rome. If the side of each arch is located at an x-intercept of the function below, determine the width of the opening of the arch, w in meters at the base.
Solution y = -(w - 6)^{2} + 16
width
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The width of the opening is the difference of the x-intercepts, which occur when y = 0.
The roots are located at 10m, and 2m. The width of the base = 10 - 2 = 8m.
Parabola Word Problem
One of the stages at a major music festival has a parabola arch with a 40' base. The stage technicians need to install a giant 30' × 17' screen at the back, but need to not let the corners be cut off from view by the parabolic arch. The arch has a maximum height of exactly 264/7 feet. Can they install the screen successfully? If so, in what direction - horizontally or vertically?
Solution
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First determine an equation for the parabolic arch. Use the maximum at (20, 264/7) and the point (0, 0). You also have the point (40, 0) but don't need it.
Try the screen horizontal, so check the point on the parabola 5' from the left-most base...
This is too low, it needs to be at least 17' for the screen to fit this way.
Try the screen vertical, so check the point on the parabola 11.5' from the left-most base...
The screen is only 30' high, so it will fit when placed vertically.
Applications of Quadratic Equations
A football player (wide receiver) has a maximum reach of 2m above the ground and can run 20m in the time a ball is thrown, starting at the same time. Given the equation, h(d) = -0.025x^{2} + 0.5x + 1.8, where h(d) is height in meters, and d is horizontal distance in meters, can the player catch the ball? Solve and show your work.
Solution
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Plug in d = 20m and compare height of the ball h(d) with the height of the receiver (2m); if the ball is less then it can be caught.
Therefore, the ball is 1.8 m off the ground at a horizontal distance of 20m. The player can reach and catch the ball.
(Side note: the x-intercept of 23.1m is not used to determine if the ball can be caught because the player doesn't have to catch the ball at the ground. Another way to answer the question is to plug in 2.0m into the height to calculate the distance.)
Word Problems
Solve the following problems.
The product of two positive, consecutive numbers is 1806. What are the numbers?
Solution
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Consecutive numbers can be written in terms of x as: x, and (x + 1)
x = +42, -43
Since the numbers are positive, then x = +42
Then using the equation for the second number (x + 1), the other number is +43.
The sum of the squares of two positive, consecutive integers is 481. Find the integers.
Solution
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Consecutive integers can be written in terms of x as: x, and (x + 1)
Using x = +16,
Then the two numbers are +16, + 17
A rectangle has a perimeter of 34 cm. Its area is 60 cm^{2}. Sketch and determine the dimensions of the rectangle using algebra.
Solution
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Solve the system with two equations. Perimeter = 2x + 2y
Area = (x)(y)
x = 15, 5 cm ... (sketch not shown)
A really wide moat surrounds, and protects a castle. The castle has a rectangular footprint measuring 100 m by 70 m. The moat is an equal width all around. Determine the width of the moat to the nearest meter, if the sum of the area of the castle and the moat together is 90,000 m^{2}.
Solution
width =
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Write the expressions for the outside width and the outside length of the moat.
The outside width = 70 + 2x
The outside length = 100 + 2x
Solve for x using the quadratic equation: a = 4, b = 340, c = -83,000
The width of the moat is 107m.
(outside width = 284m, outside length = 314m)
A matte border of uniform width is to be placed around an Italian Renaissance painting with unknown width and length. The area of the matte border is to be twice the area of the painting. If the painting is 20 cm by 30 cm, then how wide should the border be?
Solution
width =
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Write the expressions for the outside width and the outside length of the matte border.
The outside width = 20 + x
The outside length = 30 + x
Area of the border = 2(Area of the painting)
... use quadratic formula
x = 17.7, and -67.7
The width of the border should be 17.7 cm.
In a race, the Turtle is given a generous 30 minute head start. If the Turtle runs at 2 kilometers/hour, and the Rabbit runs at 8 kilometers/hour, then how many minutes after the Rabbit starts running will it catch up to the Turtle?
Solution
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minutes
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Let 't' be the time the rabbit does the race.
Turtle
Rabbit
Distance
same distance, d
same distance, d
Speed
2 km/h
8 km/h
Time
t + 0.5
t
Set the distances equal...
The rabbit will catch up to the turtle in ⅙ hour, or 10 minutes.
The floor in a sports arena is 20 m by 40 m. If a 144 m^{2} section is to be removed along with an adjacent square piece, determine the possible dimensions of the square piece.
Solution
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Let the small square sides be 'x' by 'x'.
For the small known area of 144 m^{2}, see that the side lengths are then (40 - x) and (x)...
The two possible side lengths for 'x' are 4m or 36m.
Solve for the unknown side lengths, show your work.
Solution
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Use the pythagorean theorem.
Use the pythagorean theorem. The hypotenuse (longest side) is 'c'.
Solving Right Triangles
A right triangle has the following dimensions:
Base = x Height = x + 7 Hypotenuse = 17
Substitute the side lengths into the pythagorean theorem and simplify the equation into standard form. Don't include spaces in your answer.
Solution Video
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Factor the equation above to determine the possible length of the base, 'x'.
Solution Video
x =
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x = -15, and x = +8
The side length cannot be negative, therefore x = 8
(Review) Cross Multiplication and Other Operations
Given , determine the other two primary trig ratios. Use exact values, with positive lengths.
Solution
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First use Pythagorean Theorem to find the missing length. Since , the hypotenuse = 2, and one side is 1...
Use positive root,
Solving with Trig
Calculate angle 'n' if the area of the following triangle is 50 cm^{2}.
Solution Video
n =
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degrees
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First find the height, h with the area of a triangle:
Then solve for the angle, n using trig:
Solving with Trig
A heavy rain is being blown by strong winds at an angle of 25˚ 'from the vertical'. If the rain is directly blown (at a right angle when looking above) into a 15' bridge, determine the maximum distance under the bridge that the rain reaches.
Solution
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feet
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With a quick sketch (or a strong visual imagination) you can see that the right angle triangle has a 15' height, adjacent to the 25˚ angle. The distance 'into' the bridge is the base of the right angle triangle...
Solving with Trig
Determine the angle that the line y = ½x - 3 makes with the x-axis.
Solution Video
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degrees
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Determine the side lengths of the right triangle and use trig ratios to solve...
(We could also just use the slope, 1/2 for the side lengths, and 2.)
Trig Word Problem
A very tall ladder is placed 20 m from the base of a building of unknown height, at an angle of elevation of 60˚ to the top of the building.
Determine the minimum length of ladder that must be used to get to the top of the building.
Solution Video
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Angle = 60˚
Adjacent side length = 20 m
Solve for the ladder length, which is the hypotenuse:
∴ The ladder is 40 m in length. Wowzas
How tall is the building? (Round your answer to the nearest decimal.)
Solution Video
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The building height is opposite the 60˚ angle.
∴ The building is 34.6 m high.
Trig Word Problem
If the Cobain building is 20 m tall and the Statham building is 35 m tall, and the angle of depression from the top of Cobain to the bottom of Statham is 33˚, then determine the distance from the top of Cobain to the top of Statham.
Solution Video
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First find the horizontal distance between the buildings. Draw the slanted line from the top of Cobain to the bottom of Statham.
Then find the distance (hypotenuse) between the tops...
Trig Word Problem
Gold coins are at the bottom of a diving pool, which is 14 m deep. A diver is underwater on the bottom of the pool, 20 m away from the coins and a swimmer is at the edge of the pool. The angle of depression from the swimmer to the coins is 45 degrees.
Assuming the swimmer and diver travel at the same speed (and they start going at the same time) determine the distance of the person who will get to the coins first.
Solution Video
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Compare the direct distance between the swimmer (unknown hypotenuse) and the diver (20m) from the coins.
Make a triangle with angle 45˚, depth 14 m, and the direct distance of the swimmer, the hypotenuse:
∴ The swimmer is 19.8m, and the diver is 20m from the coins, since they travel at the same speed, the swimmer is closer to the gold coins so they will get to the coins first.
If the diver can travel 4.1 m/s, and the swimmer can swim 4 m/s, determine who will get to the coins first. Show your work on paper. (distance = speed × time) [1]
Solution Video
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Solve for t and compare using the d-s-t formula...
Time of the swimmer:
Time of the diver:
∴ The diver gets to the gold coins first.
Trig Word Problem
Use a variable to determine the height, h where distance is in meters. Round your answer to the nearest decimal place.
Solution Video
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Separate the big scalene triangle into 2 smaller right angle triangles.
Let x be the base on the left. Make an equation for the left side using tan.
Let 30 - x be the base on the right. Make an equation for the right side.
Solve the system of equations: set the two h equations equal and solve for x:
Substitute x into any one of the equations to solve for h:
Trig Word Problem
Some people are trying to measure the height of a super-tall statue. Point B is S 45˚ W from the bottom of the statue, and point C is E 45˚ S from the bottom. Let point A be the top of the statue and point D the bottom. ∆ABD and ∆ACD are in the vertical plane while ∆BDC is in the horizontal plane. The distance between point B and C is measured at 68 m.
If the angle of elevation from point B to the top is 24˚ and from point C to the top is 38˚, solve for DC Solution
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Solve the system of equations with 2 unknowns (variables): BD & DC...
Sub ② in ①...
Find another triangle where you can make an equation that only has BD and DC in it...
Sine law is used to solve acute and obtuse triangles, which you otherwise couldn't using S^{OH}C^{AH}T^{OA}.
Solving for the angles use...
Lowercase represents side lengths (a, b, c, ...)
Uppercase represents angles (A˚, B˚, C˚, ...)
Use any two of the fractions/ratios
Solving for the side lengths use...
Notice it's just the same as before, but reciprocated (flipped).
When do you use Sine law?
Use Sine law any time you have this pair: 1 side length opposite 1 angle.
You need 3 values to solve for the 4^{th} unknown.
Some examples where you can solve for the red variable...
Cosine Law (for Acute and Obtuse Triangles)
Cosine law is used to solve acute and obtuse triangles, which you otherwise couldn't using S^{OH}C^{AH}T^{OA}.
Solving for the side lengths use...
Solving for the angles use...
Notice you're just rearranging the previous equations, isolating for the cos X˚... It's simple order of operations.
When do you use Cosine Law
Any time you have 3 values, then you can solve for the 4^{th} unknown.
Some examples where you can solve for the red variable...
You must use the quadratic equation when solving for these things...
Sine and Cosine Law
Which of the following would be used to solve for the first missing value on the triangle below?
Solution
There is an opposite angle-side pair here: 65˚ & 15.
Therefore use the Sine law first.
Which of the following would be used first, to solve the triangle.
Solution
Use COSINE LAW when you have adjacent Side-Angle-Side (SAS), or Side-Side-Side (SSS).
In this case you have SAS so use:
Word Problems with Sine and Cosine Law
Daniel, a scuba diver, swims underwater 300 m north from the dive boat. He then turns right 120˚ and swims 400 m. How horizontally far from the boat is he, to the nearest meter? Round to the nearest tenth decimal. (A diagram is given here, but on a school test it might not be given so you would draw it.)
Solution
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Use cosine law
Sine and Cosine Law
An obtuse triangle has side lengths 10cm, 5cm, and 6cm.
Determine any one of the angles of the triangle. Round to the nearest tenth decimal place. [1]
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An obtuse triangle is a triangle with one angle greater than 90˚ (an acute triangle is a triangle with all angles less than 90˚).
The other angles are approximately 22.3˚ and 27.1˚
You have side lengths and angles,
First determine the height with a right angle triangle, using the one on the right-hand side here...
Then calculate the area,
Word Problems with Sine and Cosine Law
From the top of a cliff, the angle of elevation to an airplane in the sky is 20˚. The angle of depression to a boat, directly under the plane, is 40˚. The distance from the boat to the top of the cliff is given. Find how high the plane is above the boat.
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km
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Find the angle of the part of the smaller, upper triangle at the plane:
The total angle at the cliff is 60˚ (20˚ + 40˚).
Use the sine law to find the height of the plane above the boat using the whole, large triangle (let C equal the distance PB from the plane to the boat):
Word Problems with Sine and Cosine Law
Two buildings are 40 m apart. David and Helen work in adjacent office buildings. Use the given information to determine the height of Helen’s window. The angle of depression from David’s window to the base of Helen’s building is 33 degrees. The angle of elevation from David’s window to Helen’s window is 20 degrees.
Solution
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m
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The following is just one way of solving it using the sine law, you may get the right answer with various other approaches (including right triangles):
The total angle at David is 53˚ (20˚ + 33˚).
Calculate the distance from David to the ground:
Calculate the angle at Helen's window
Calculate
Word Problems with Sine and Cosine Law
Solve for length, AB. (Round your answer to the nearest tenth of a meter).
Solution Video
AB =
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m
Hint Unavailable
Calculate angle ANB:
Calculate length BN:
Then calculate length AB using the sine law: