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# Principles of Math MPM2D

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

# Analytic Geometry

## Given the equation for a circle with a center at (h, k)...

\begin{align} (x - h)^2 + (y - k)^2 = r^2 \\ \\ \end{align}

## 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.

\begin{align} (x - 10)^2 + (y - 14)^2 = 3^2 \\ \\ \end{align} # Geometric Properties

## 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)

A (5, 9)
B (-10, -9)
C (15, -10)

# Quadratic Relations

xa × xb = xa + b

xa ÷ xb = xa - b

(xa)b = xa × b

## Simplify fully to an integer or fraction using positive exponents, showing your work without using a calculator.

$a^{-n} = \left( \cfrac{1}{a^{\ +n}} \right) \quad\quad or \quad\quad \left( \cfrac{a}{b} \right)^{\!\!-n} = \left( \cfrac{b}{a} \right)^{\!\!+n}$

## Solve the following exponential equations, similar to the steps shown in the example below. Simplify fully.

\begin{align} 3^{(3x + 2)} & = 243 \\ 3^{(3x + 2)} & = 3^{(5)} \\ 3x + 2 & = 5 \\ 3x & = 5 - 2 \\ 3x & = 3 \\ x & = 1 \\ \\ \end{align}

## The table below shows the flight path of a ball, with distance and height in meters.

 Distance (x) Height (y) 0 1 1 8 2 13 3 16 4 17 5 16 6 13 7 8 8 1

## 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.

$h(t) = -7.3t^2 + 8.25t + 2.1$

## Given the general, vertex form of a quadratic.

$y = ±\,a(x - h)^2 + k$

## Given the general form of a quadratic, determine the equation of a quadratic function with the transformations listed below....

$y = a(x - h)^2 + k$

## Based on the value 'n' determine some of the horizontal transformations below.

$y = a\left[ n\left(x - h\right)\right]^2 + k$

## Find the vertex using each of the following methods: # Quadratic Expressions

Specific Topic General Topic School Date
The Correct Way to Expand a Squared Binomial Expanding Branksome Sep 2013
Factoring a Perfect Square Trinomial Quadratics Branksome Sep 2013
Factoring Unordered Trinomials Example 1 Quadratics Branksome Sep 2013
Factoring Unordered Trinomials Example 2 Quadratics Branksome Sep 2013

## 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.

$6x^2 + 11x + 3$ What multiplies to +18 and adds to +11? = +9 and +2 \begin{align} & = (2x \quad\quad )(3x \quad\quad ) \\ \\ & = ( \ \underbrace{2x + \underbrace{3 \ )( \ 3x}_{+9} + 1 }_{+2} \ ) \\ \\ \end{align}

## Factor the following perfect squaresfully, using any method (direct is encouraged). The general formulas for perfect squares are given below.

\begin{align} & a^2 + 2ab + b^2 \\ \\ & = (a)^2 + 2ab + (b)^2 \\ \\ & = (a + b)^2 \\ \\ \end{align} \begin{align} & a^2 - 2ab + b^2 \\ \\ & = (a)^2 - 2ab + (b)^2 \\ \\ & = (a - b)^2 \\ \\ \end{align}

## Given the general formula for a perfect square quadratic...

E.g. 1) (a)2x2 + 2(a)(b)x + (b)2
= (ax + b)2
E.g. 2) x2 + 6x + 9
= (a)2x2 + 2(a)(b)x + (b)2
= (1)2x2 + 2(1)(3)x + (3)2
= (1x + 3)2

## Factor the difference of squares fully (meaning common factor whenever possible). Show as much work in your notes as you need.

\begin{align} & 1x^2 - 9 \\ \\ & = (1x + 3)(1x - 3) \\ \\ \end{align} \begin{align} & 75a^2 - 27 \\ \\ & = 3(25a^2 - 9) \\ \\ & = 3\left[ (5a)^2 - (3)^2 \right] \\ \\ & = 3(5a + 3)(5a - 3) \\ \\ \end{align}

## 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

# Quadratic Equations

Specific Topic General Topic School Date
Solving for X by Completing the Square Factoring Quadratics Branksome Sep 2013

## Given the equation already in factored form.

\begin{align} y = x(2x - 3) \\ \\ \end{align}

## Determine the solutions, hence solve. Use the zero-product property or complete the square where most applicable. Order solutions from lowest to highest... ## The revenue, expense, and profit functions are given below, where x is the number of items sold.

\begin{align} & Revenue = 5x^2 + 200 \\ \\ & Expense = 35x^2 - 130x + 100 \\ \\ & Profit = Revenue \ - \ Expense \end{align}

## 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) = 2t2 - 16t + 682

# Trigonometry with Right Triangles

## The interior angles of a triangle are given in terms of x in the diagram below. Base = x
Height = x + 7
Hypotenuse = 17

## 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. ## 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. ## 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. # Trigonometry with Acute and Obtuse

## Given the complex shape, determine the missing quantities stated below. ### Explain how the Cosine law is based on the Pythagorean theorem.  Solution Hint Clear Info Incorrect Attempts: CHECK Hint Unavailable Pythagorean Theorem $a^2 + b^2 = c^2$ Cosine law when angle (A˚) = 90˚ \begin{align} a^2 & = b^2 + c^2 - 2bc \cos A˚ \\ \\ a^2 & = b^2 + c^2 - 2bc \cos (90˚) \\ \\ a^2 & = b^2 + c^2 - 2bc \cancelto{0}{\cos (90˚)} \\ \\ a^2 & = b^2 + c^2 - \cancelto{0}{2bc (0)} \\ \\ a^2 & = b^2 + c^2 \\ \\ \end{align} The Cosine law is the Pythagorean theorem when the angle is 90˚... Loading and rendering MathJax, please wait...
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