Terms You Need to Know and Understand

MPM 2DILinear SystemsDay 1-1

Terms you need to know and understand:

Slope -

A System of Equations is two or more equations considered together.

Formal Check Left Side (LS) and Right Side (RS) are calculated separately. If a point (x , y) or a set of coordinates satisfies an equation then that point is a solution and LS = RS. If a point satisfies an equation, then the point is on the line.

Solve by Substitution

To solve a system of two linear equations with two variables

1. Solve one equation for one of its variables
2. Substitute the expression from step 1 in the other equation and solve for the remaining variable
3. Substitute the value of the variable from step 2 into one of the original equations to find the value of the second variable
4. Check the solution in each of the original equations

During the process of substitution you may reach a line such as:

0y = 0OR0x = 2

If the equation is true for all values of the variable, then the two lines coincide and there are infinitely may solutions.

If the equation is never true for any value of the variable, then the lines are parallel and distinct, so there are no solutions.

Solve by Elimination

To solve a linear system in two variables by elimination

1. Clear decimals and fractions, if necessary
2. Rewrite the equations with like terms in the same column
3. Multiply one or both equations so that the coefficients of one variable are the same or opposites
4. Add or subtract the equations to eliminate a variable (add or subtract to get zero) and solve the resulting equation for the remaining variable
5. Substitute the value from step 4 into one of the original equations and solve for the other variable
6. Check the solution in each of the original equations

Examples

1.Solve for x

a)x + 7y = 9b) 7y – 2x = 3

1. Solve by substitution. If there is exactly one solution, check your answer.

a)x – 4y = 8b) x – 2y = 8

2x – 8y = 82x – 8y = 8

1. Solve by elimination. If there is exactly one solution, check your answer.

a)4x – 3y = 5

8x – 6y = 10

b)

Homework

Worksheet 1.3 #1, 2adg, 3, 4, 5

Worksheet 1.5 #1adegjklm, 2abd, 3

MPM 2DILinear SystemsDay 2-1

Review – Graphing a Linear Equation

Method 1 – Table of Values4y – x + 8 = 0

MPM 2DILinear SystemsDay 2-1

MPM 2DILinear SystemsDay 2-1

Method 2 – Slope y-intercept Method

MPM 2DILinear SystemsDay 2-1

y = mx + b

m = b =

-4x + y + 5 = 0

MPM 2DILinear SystemsDay 2-1

Method 3 – Two Intercept Method

MPM 2DILinear SystemsDay 2-1

2x – 3y = -6

x-intercept

y – intercept

MPM 2DILinear SystemsDay 2-1

Solving a Linear System by Graphing

1. Look at the equations and determine which method is the easiest / least time consuming to use.
2. Identify any values you will be using to graph each line.
3. Graph each line.
4. Identify the point of intersection, if there is one.
5. Make your conclusion.

Example: Solve by graphing.

-x + 2y – 2 = 03x – 2y – 10 = 0

Graphical Analysis of the 3 Possible Outcomes

Word Problems

General Strategy:

Read the question carefully – read it a second or third time if necessary

Look for key words that can be easily translated into math symbols

Use a chart to organize the information if one applies

Identify the type of question – number, distance-speed-time, mixture, measurement

Identify and define your two variables

Create your two equations – generally each sentence forms one equation

Solve your linear system

Write a concluding statement

The sum of two numbers is 255. When the smaller is subtracted from the larger, the result is 39. Find the numbers.

Erica drove from Sarnia at 80 km/h. Aisha left Sarnia one hour later and drove along the same road at 100 km/h. How far from Sarnia did Aisha overtake Erica?

Malcolm has been asked to produce 500 mL of a 62% acid solution. He has been given two beakers, one containing an 80% solution, the other containing a 50% solution. How much of each type of solution should he use?

Homework

Worksheet 1.2 #1abefj, 2, 3, 4

Worksheet 1.7 #1-3, 5-8

MPM 2DIAnalytic GeometryDay 4-1

Length of a Line Segment (What is the distance between two points?)

Length =

=

Determine the length of AB.

What is a perfect square?

List the first 15 perfect squares.

Use the formula to determine the distance between P (-7, 3) and Q (-1, -4). Express your answer as an exact solution and as an approximate solution to the nearest tenth.

Equation of a Circle (Centred on the origin)

1.What is the equation of the circle, on the origin, with a radius of 7 cm?

2.A circle has the equation x2 + y2 = 121, what is the length of the radius?

Midpoint of a Line Segment

Since the Midpoint of a Line Segment is a point, it MUST be expressed as a point  with brackets and a comma to identify the coordinates.

Midpoint of a Line Segment =

Determine the slope and midpoint of the following line segments.

Horizontal Line Vertical Line

Diagonal Line

Example

If A (2, 3) is the endpoint of a line segment and M (-1, -3) is the midpoint of the line segment, what are the coordinates of B, the other endpoint?

Equations of Special Lines

Horizontal Lines – the value of the y coordinate is always the same, so the equation is y = k, where k is the y-coordinate of any point on the line. The slope of a horizontal line is zero.

Vertical Lines – the value of the x coordinate is always the same, so the equation is x = h, where h is the x-coordinate of any point on the line. The slope of a vertical line is undefined.

Equation Format

Standard Form:Ax + By + C = 0A, B, and C must be integers, A must be positive

Slope Intercept Form:y = mx + b

Slope Point Form:y – y1 = m(x – x1)(x1, y1) is the given point

y = m(x – p) + q(p, q) is the given point

Determine the Equation of a Line

Given: slope and one point

• Substitute the slope for m
• Substitute the coordinates of the point
• Simplify the equation

Example: What is the equation of the line with slope –1/3 passing through (2, 5)?

Given: two points

• Calculate the slope
• Substitute the slope for m
• Substitute the coordinates of one of the points
• Simplify the equation

Example: What is the equation of the line, in Standard Form, passing through (-1, 7) and (6, -3)?

Parallel and Perpendicular Lines

Parallel Lines – have the same slope

Perpendicular Lines – the slopes are negative reciprocals. (If you multiply them the product is –1).

Identify the perpendicular slope for each of the given slopes.

m = 2, m = m = , m = m =, m =

Examples:

a)What is the equation of the line passing through (4, 6) andparallel to y = -2x + 4?

b)What is the equation of the line passing through (4, 6) andperpendicular to y = -2x + 4?

Homework

Worksheet 2.1 #1a, 2, 3, 5, 7, 8, 9

Worksheet 2.3 #1abdf, 3, 4, 5, 6, 7

Worksheet ABC Practice – A#1,2, B#1e-i, 2bdfh, C#1acegi, 2bdf, 3d-h

MPM 2DIAnalytic GeometryDay 5 - 1

Definitions

Median: a line joining a vertex to the midpoint of the opposite side

Altitude: a perpendicular line joining a vertex to the opposite side

Perpendicular Bisector: a perpendicular line that cuts through the midpoint of a line segment.

Diagonal: a line drawn from two non-adjacent vertices.

Everything that you will be asked to verify will require 1 or more of the properties (formulas) that we’ve just reviewed.

Identify the following quadrilaterals and mark the key features on the diagram (ie equal lengths, parallel sides,…)

Draw an example of an altitude, median, and a perpendicular bisector using the given triangles.
Examples

Triangle ABC has vertices A(3, 4), B(-5, 2), and C(1, -4). Determine the equation for:

a)CD, the median from C to AB

b)AE, the altitude from A to BC

c)GH, the right bisector of AC

A quadrilateral has vertices P(-3, 1), Q(-1, -5), R(11, 1) and S(1, 3). Verify that:

a)quadrilateral PQRS is a trapezoid

b)the line segment joining the midpoints of the two non-parallel sides is parallel to the base

c)the line segment from (b) is half as long as the sum of the lengths of the top and bottom

Distance from a Point to a Line

The shortest distance from a point to a line is the perpendicular distance.

To calculate the shortest distance:

1. find the slope of the given line
2. determine the slope of the perpendicular line
3. find the equation of the line perpendicular to the given line
4. find the point of intersection of the given line and the perpendicular line
5. find the length (distance) using the point of intersection and the given point

Example: Find the distance from P(4, 1) to the line joining Q(1, -8) and R(-4, 2) as an exact value and to the nearest tenth.

Let X represent the point on QR that will give you the shortest distance.

1. Find the slope of QR. (Find the equation of QR since it wasn’t given)

1. Identify the slope of the perpendicular line PX.

1. Find the equation of the perpendicular line PX, using P and the slope from step 2.

1. Find the point of intersection of PX and QR.

1. Find the length of PX.

MPM 2DIPolynomialsDay 6 - 1

Like Terms – terms that have the same variable(s) and exponent(s)

Coefficient – the number in front of the variable (numerical coefficients)

Constant – a number by itself; a value that doesn’t change

Perfect Square – a number that produces an integer result when square rooted. The result when multiplying a number (expression) with itself.

1.Simplifying Expressions

Instructions will be either Expand or Expand and Simplify

Collect like terms (group them together)

Add or subtract the coefficients of like terms

Read the signs on each term carefully

**Exponents change ONLY when you multiply or divide terms.

SimplifyExpand and Simplify

(2m – n) – (3m + 4n)2(x2 – 3x – 4) – 3(4x2 – x + 5)

2.Multiplying Binomials

Every term in the first bracket gets multiplied with each term in the second bracket (some teachers refer to this as FOIL)

Arrange the terms in descending order (by exponents)

Drawing in arrows will help you keep track of what terms you’ve multiplied (rainbow)

Expand

a)(2x + 1)(x – 7)b)(4x – 5y)(2x + 3y)

c)(x + 3) – 2(x – 1)(x + 2)d)(x + 2)( x2 + 3x – 1)

Special Products

Sum and Product

Perfect Square Trinomials

Difference of Squares

**Recognizing the patterns that develop from expanding these binomials will help you factor.

A – Sum and Product: x2 + bx + c

Expand

(x + 7)(x + 2)(x – 5)(x – 3)(x – 4)(x + 1)

**This is referred to as a sum and product because the coefficient of the second term (b) is the sum of the constants in the binomial and the constant term (c) is the product of the constant terms in the binomial.

B – Perfect Square Trinomials: ax2 + bx + c, a and c are perfect squares.

Expand

(x + 5)2(x – 3y)2(5y – 2x)2

**Square the first, square the last, twice the product of the first and last – this rhyme may help you square a binomial to get a perfect square trinomial more quickly. Be careful if you use it that you are using it properly.

C – Difference of Squares: a2 – b2, a and b are perfect squares

Expand

(x + 1)(x – 1)(2x – 5)(2x + 5)(7p – 3q)(7p + 3q)

**Notice that only two terms remain, the operation between them is a minus sign (difference) and each of the terms is a perfect square, hence, difference of squares.

Expand and simplify.

2(x + 1)(2x + 3)

= 2[x(2x) + x(3) + 1(2x) + 1(3)]expand the binomials first (optional line)

= 2[2x2 + 3x + x + 3]multiply

= 2[2x2 + 4x + 3]simplify the like terms

= 2(2x2 ) + 2(4x) + 2(3)expand the constant (optional line)

= 4x2 + 8x + 6multiply

Expand and simplify.

3t2 – (3 – 2t)2 + 5(2t – 1)(2t + 1)

2(3m – n)2 – 3(2m – 5)(m + 3)

Homework:

Worksheet 3.1 #4aceg, 5aceg, 7, 9

Worksheet 3.2 #2, 3, 5

Worksheet 3.3 #1bdfhj, 2, 4, 5

MPM 2DIPolynomialsDay 7 - 1

Common Factorthe polynomial can have two (2) or more terms

look for the largest number and/or variable that can be removed from every term in the polynomial without getting negative exponents

Examples

a)4x + 8b)6x3y2 – 12x2y3 + 8x4yc)7x3y2 – 4x5y7

d)5(x – 3y) – 2x(x – 3y)

Sum and ProductMUST be a trinomial (3 terms)

a=1 in the expression ax2 + bx + c

b is the sum of the two numbers in the binomial factors

c is the product of the two numbers in the binomial factors

IF the sign in front of ‘c’ is positive, then the signs in the binomials are the same and the sign in front of ‘b’ tells you if they are both positive or both negative

IF the sign in front of ‘c’ is negative, then the signs in the binomials are different and the sign in front of ‘b’ tells you if the larger number is positive or negative

Examples

a)x2 + 11x + 24b)x2 – 11x + 24c)x2 + 5x – 24d)x2 – 5x – 24

Difference of SquaresMUST be a binomial (2 terms)

it MUST have a minus sign

the two terms MUST be perfect squares

the square roots of the terms are the two terms for the binomial

one of the binomials will have a minus sign, the other a plus sign

Examples:

a)x2 – 36b)4x2 - 25

Factor by Groupingthe polynomial USUALLY has 4 terms

rearrange the terms if necessary so that pairs of terms have something in common

factor the pairs, then check to see if the resulting binomials are common

Example

a)y2 – xy – x + yb)x2 – 2xy + 5x – 10y

DecompositionMUST be a trinomial (3 terms)

a≠1 in the expression ax2 + bx + c

the product is a  c, the sum is b

make a list of the factors of the product, use the pair that gives you the appropriate sum (decompose the middle term - then factor by grouping)

Example

a)8x2 + 22x + 15b)3x2 + 16x – 12

ALWAYS look for a common factor first – this will make the numbers you’re working with more manageable most of the time.

Note: If the binomials are the same, write your answer as a binomial squared.

Remember:You can always check your answer by expanding

More Examples

Factor completely, if possible.

a)x2 – x – 30b)y2 – 20y + 36c)x3 + 18x2 + 72x

d)ax2 + 10ax – 24ae)2x2 + 7x – 15f)4x2 – 14xy + 6y2

g)6x2 – xy – 35y2h)x2 + 1i)5x2 – 45

j)x4 – 16k)x16 – 1

MPM 2DIQuadratic FunctionsDay 9-1

Functions:

A function is a set of ordered pairs, such that, for every value of x, there is a different y value.

When given a graph you can test whether or not a relation is a function by using the vertical line test. If your vertical line passes through two points anywhere on the graph, then the relation is not a function.

Are the following functions? Explain.

a)(1,5), (2,4), (-2,5), (6,7)YES / NO

b)(7,-2), (4,1), (-5,3), (-4,2), (7,6)YES / NO

Graphing Parabolas and Identifying Key Features

Create a table of values, then graph y = x2

d)

e)What patterns do you notice?

f)

g)

h)

i)

j)

k)

l)

m)

y = x2 is a parabola. It is symmetric which means that there is a line (axis of symmetry) that can be used to fold the graph and have both halves match perfectly.

Transformations:

A translation is a shift in the graph in any direction. The shape of the graph is not changed.

A dilatation is an enlargement (stretch) or reduction (compression) depending on the stretch factor.

A reflection produces a mirror image of the graph. It ‘flips’ the points to the other side of the axis of symmetry.

Summary

When a function is written in the form y = a(x – h)2 + k, the values of a, h and k are responsible for a number of different transformations.

a is responsible for reflections, stretches and compressions.

If a > 0, then the graph opens

If a < 0, then the graph opens

If a >1, then there is a

If 0< a < 1, then there is a

h is responsible for horizontal translations.

If h > 0, then the graph moves

If h < 0, then the graph moves

k is responsible for vertical translations.

If k > 0, then the graph moves

If k < 0, then the graph moves

The vertex of any parabola is ( , ).

If you can identify the vertex, then the axis of symmetry will be x =

The direction of opening is determined by the value of .

There will be a maximum value if and a minimum value if . The graph has a max/min value of y = when x = .

The stretch factor is determined by the value of and will change the step pattern, which can be used to plot the different points (and draw the graph) after determining the vertex.

The range is affected by the value of .

The y-intercept can be found by .

The number of x-intercepts depends on the values of a and k.

If a > 0 and k > 0, there will be x-intercepts (zeros).

If a > 0 and k < 0, there will be x-intercepts (zeros).

If a < 0 and k > 0, there will be x-intercepts (zeros).

If a < 0 and k < 0, there will be x-intercepts (zeros).

y = ax2

n)

y = ax2 + k

y = a(x – h)2 + k

Memorize the vertex form for the equation of a parabola – make special note of the signs in the formula. This will help you correctly identify the values of a, h, k and any key feature.

Identify the vertex for each of the following functions. Determine the number of x-intercepts for each function.

a)y = 2(x + 3)2 – 5vertex ( , )# of intercepts =

b)y = -(x – 1)2 + 2vertex ( , )# of intercepts =

c)y = -3(x – 9)2 – 11vertex ( , )# of intercepts =

d)y = 4(x + 6)2 + 7vertex ( , )# of intercepts =

MPM 2DIQuadratic FunctionsDay 10-1

Completing the Square

Recall (x – 7)2

= x2 – 14x + 49

If you were given only x2 – 14x and asked to determine the binomial that would make this a perfect square trinomial, how could you have determined that it was (x – 7)2? In other words, how are b and c related to each other and to the value (-7) in the binomial?

b = -14

c = 49

Identify the value, c, that completes this trinomial; then identify the binomial factor.

x2 + 8x + = ( x + )2

x2 – 12x + = (x – )2

x2 + 7x + = (x + )2

Convert from standard form y = ax2 + bx + c to vertex form y = a(x – h)2 + k by completing the square.

Steps:

1.Factor any ‘a’ value from the first two terms of the trinomial.

2.Use the two terms you’ve just factored (inside the brackets) to make a perfect square trinomial  complete the square.

3.Clean up the equation.

Complete the square to determine the maximum or minimum value of the following function.

y = -2x2 + 20x – 44

the vertex is ( , ) and the function has a value of y = when x = .

Determine the maximum or minimum value of the following function.

y = 3x2 – 18x + 13

the vertex is ( , ) and the function has a value of y = when x = .

Solving Word Problems

The word problems you will be solving in this unit will require you to find either the maximum or minimum possible value given certain constraints.

1.Is a formula required? (perimeter, area, …)

2.Define your variables.

3.Determine your equations. One is probably linear, the other quadratic.

4.Solve the linear equation for one of the variables (y = ….)

5.Substitute into the quadratic equation.

6.Expand and simplify if necessary.

7.Complete the square to determine the vertex. (This is your max/min value and when it occurs)

8.Reread the questions to make sure you’ve solved the correct problem.

9.Make your conclusions.

Word Problems

Find two numbers whose sum is 156 and whose product is a maximum.

the vertex is ( , ). The maximum product is , it occurs when . The other number is .