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MPM 2D Course Review Unit 1: Linear Systems

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**I can graph a line 3 ways: Table of values x- and y-intercepts**

y-intercept and slope Ex. 1) y=-3x+2

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**I can graph a line 3 ways: Table of values x- and y-intercepts**

y-intercept and slope Ex. 2) 4x-3y=12

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**I can graph a line 3 ways: Table of values x- and y-intercepts**

y-intercept and slope Ex. 3) y=(2/3)x-5

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**I can translate a word problem into algebra**

What important, defining information are you missing? THIS WILL HELP YOU DEFINE YOUR VARIABLES Hint: The question at the end will direct you to at least one of the variables EXAMPLE: Karl owns a small airplane. He pays $50/h for flying time and $300/month for hangar fees at the local airport. If Karl rented the same type of airplane at a flying club, it would cost him $100/h. When will the monthly cost of owning and renting be the same?

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**I can solve a linear system by graphing**

Karl’s situation: y1=50x+300 y2=100x

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**I can determine if a point is the solution to the system**

…using substitution! We found the POI for Karl was (6, 600) on the graph. How do we test our answer? Plug it into both equations! y1=50x+300 y2=100x

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**I can determine the number of solutions by looking at the equations in the system**

Different slope No solutions Same slope, different y-intercepts Infinite solutions same y-intercept

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**I can solve a linear system by substitution**

2𝑥−𝑦=2 and 3𝑥−3𝑦=−3 CHECK YOUR ANSWER!

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**I can solve a linear system by elimination**

2𝑥−𝑦=2 and 3𝑥−3𝑦=−3 CHECK YOUR ANSWER!

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**I can solve problems using linear systems**

A weekend at a lodge costs $360 and includes 2 nights’ accommodation and 2 meals a day. A week costs $1200 and includes 7 nights’ accommodation and 10 meals. What is the cost of one night and one meal? How much would it cost for five nights and 4 meals? CHECK YOUR ANSWER!

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**Unit 2: Analytic Geometry**

MPM 2D Course Review Unit 2: Analytic Geometry

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**I can calculate the length of a line segment, given its endpoints**

Distance formula: 𝑑= 𝑥 2 − 𝑥 𝑦 2 − 𝑦 1 2 Find the distance between (-7, 1) and (5, -2)

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**I can find the midpoint of a line segment**

Midpoint formula: 𝑀=( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 ) Ex 1. Find the midpoint between (-4, 6) and (8, -2)

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**I can find the midpoint of a line segment**

Midpoint formula: 𝑀=( 𝑥 1 + 𝑥 2 2 , 𝑦 1 + 𝑦 2 2 ) Ex 2. Find the other endpoint of a line segment if one endpoint is (-4, -2) and the midpoint of the line segment is (2, 6).

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**I can solve problems involving midpoints, medians, and perpendicular bisectors**

Median: a line that joins a vertex of a triangle to the midpoint of the opposite side. Perpendicular Bisector: of a line segment is the line that is perpendicular to the line segment and passes through the midpoint of the line segment.

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**I can solve problems involving midpoints, medians, and perpendicular bisectors CONT’D**

STEPS TO FIND EQN OF MEDIAN: Calculate the midpoint of the line opposite the vertex of interest Calculate the slope of the line connecting the vertex of interest to that midpoint (ie. The median) Sub the slope into the eqn for a line: y=mx+b Plug in (x, y) (either the midpoint or the vertex of interest) Solve for b State equation with slope (m) and y-intercept (b) Find the equation of the median line from vertex C in triangle ABC if the coordinates of the vertices are A(-3, 3), B(2, -5), and C(5, 2)

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**I can solve problems involving midpoints, medians, and perpendicular bisectors CONT’D**

STEPS TO FIND EQN OF PERPENDICULAR BISECTOR: Calculate the midpoint of the line segment Calculate the slope of the line segment The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment it is bisecting Plug the slope and the midpoint into y=mx+b Solve for b State equation with slope (m) and y-intercept (b) Find the equation of the perpendicular bisector of the line segment from A(1, 1) to B(5, 3)

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**I can classify triangles given the coordinates of the vertices**

The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Determine what kind of triangle it is.

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**I can classify triangles given the coordinates of the vertices**

The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Determine what kind of triangle it is.

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**I can verify properties of geometric figures algebraically**

The vertices of triangle ABC are A(5, 5), B(-3, -1), and C(1, -3). Show that the median from vertex C is half as long as the hypotenuse.

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**I can verify properties of geometric figures algebraically CONT’D**

Classify the shape with vertices at A(-4, 2), B(2, -5), C(7, -3) and D(1, 5)

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**I can verify properties of geometric figures algebraically CONT’D**

PQRS is a rhombus with vertices at P(3, 3), Q(0, 1), R(3, -1), and S(6, 1). Verify that its diagonals bisect each other at right angles.

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**I can determine the shortest distance from a point to a line**

…the shortest distance from a point to a line is always the PERPENDICULAR PATH from the point to the line.

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**I can determine the equation of a circle centered at the origin, given the circle’s radius**

𝑥 2 + 𝑦 2 = 𝑟 2 State the equation of the circle: (a) centre origin, radius of 8 (b) centre origin, radius of 1 3

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**I can determine the radius of a circle centered at the origin, given the circle’s equation**

Length of a circle’s radius: 𝑟= 𝑥 2 + 𝑦 2 A circle has the equation 𝑥 2 + 𝑦 2 =144. Where is its centre? What is its radius?

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**I can sketch a circle, given its equation**

A circle has the equation 𝑥 2 + 𝑦 2 =36. Sketch its graph.

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**I can determine the equation of a circle, given a point through which it passes**

Find the equation of the circle passing through (-5, 12)

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**STUDY Shapes and their names and properties**

Important terms (midpoint, median, perpendicular bisector) Formulas

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**Unit 3: Quadratics (Standard & Factored Form)**

MPM 2D Course Review Unit 3: Quadratics (Standard & Factored Form)

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**I can simplify expressions involving exponents**

𝑎 𝑥 × 𝑎 𝑦 = 𝑎 𝑥+𝑦 𝑎 𝑥 ÷ 𝑎 𝑦 = 𝑎 𝑥−𝑦 𝑎 𝑥 𝑦 = 𝑎 𝑥𝑦 𝑎 0 =1 𝑎 −𝑥 = 1 𝑎 𝑥 48 𝑥 3 𝑦 4 −8 𝑥 5 𝑦 3

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**I can expand & simplify polynomials**

Distributive property 𝑎(𝑏+𝑐) = 𝑎𝑏 + 𝑎𝑐 FOIL (first, outside, inside, last) 𝑥+𝑏 𝑥+𝑐 = 𝑥 2 +𝑐𝑥+𝑏𝑥+𝑏𝑐 Ex. 2 𝑥+1 𝑥−3 −(4𝑥+3)(𝑥−1)

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I can factor using GCF 28 𝑥 2 −7𝑥 5 𝑥 5 𝑦 𝑥 2 𝑦

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**I can factor differences of squares**

𝑥 2 − 𝑦 2 = 𝑥+𝑦 𝑥−𝑦 100 𝑥 2 −81 128−2 𝑚 2

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I can factor 𝑥 2 +𝑏𝑥+𝑐 𝑥 2 −29𝑥+28 𝑥 2 −4𝑥−21

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I can factor 𝑎𝑥 2 +𝑏𝑥+𝑐, 𝑎≠1 20 𝑥 2 +6𝑥−2 4𝑥 2 −16𝑥+15

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**I can solve a quadratic equation by factoring**

𝑥 2 +9𝑥+14=0 6 𝑥 2 −𝑥=15

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**I can identify the key features of a graph of a parabola**

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**I can graph a parabola (determine the zeros and vertex ) from 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐**

𝑦=− 𝑥 2 +6𝑥−9

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**I can find the equation of a quadratic given its zeros and another point on the parabola**

A parabola has zeros at (-2, 0) and (4, 0), and passes through (2, 16). Find its equation.

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**I can solve problems involving quadratics**

When Kermit the Frog makes a giant leap from one lily pad to another, he follows a parabolic path. Kermit is in the air for 6 seconds before he makes a safe landing. Kermit knows that after 2 seconds, he is 72 cm high. How high is Kermit at his greatest height?

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**I can solve problems involving quadratics**

A design engineer uses the equation ℎ=− 𝑑 to model an archway for the entrance to a fair, where h is the height in metres above the ground, and d is the horizontal distance from the centre of the arch. How wide and tall is the arch? For what values of d is the relation valid? Why? If a width of 2.5 m is needed per line-up at the entrance, how many line-ups can there be?

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Perpendicular Bisector- a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side Theorem 5.1 Any point.

Perpendicular Bisector- a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side Theorem 5.1 Any point.

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