Ch. 3 & 4 - Quadratic Functions and Equations Notes Block D · Chapter 3 & 4 – Quadratic...

Pre-Calculus 11 Chapter 3 & 4 – Quadratic Functions and Equations

Created by Ms. Lee 1 of 31 Reference: McGraw-Hill Ryerson Pre-Calculus 11

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Ch. 3 & 4 – Quadratic Functions and Equations

3.2 – PROPERTIES OF QUADRATIC FUNCTIONS 2

Ch. 3. 2 HW: p. 174 # 1 ‐ 3, 5a,b, 6a, 7, 8 4

3.1 – INVESTIGATING QUADRATIC FUNCTIONS IN VERTEX FORM 5

Ch. 3.1 HW: p. 157 #1 – 9 odd letters 9

3.3 – COMPLETING THE SQUARE 10

Ch. 3.3 HW: p. 192 #1 – 4, 6 – 8 odd letters 14

4.1 – GRAPHICAL SOLUTIONS OF QUADRATIC EQUATIONS 15

Ch. 4.1 HW: p. 215 #1, #3a,b,e, 4 ‐ 7, 10, 12 17

4.2 – FACTORING QUADRATIC EQUATIONS 18

Ch. 4.2 HW: p. 229 #1 – 10 odd letters, 11, 14, 16, 17, 19a, 20, 27 odd letters 20

4.3 – SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE 21

Ch. 4.3 HW: p. 240 #2, 4, 5, 6 odd letters 22

4.4 – QUADRATIC FORMULA 23

Ch. 4.4 HW: p. 254 #3 – 5 odd letters, 13, 14, 18 24

4.4 – THE NATURE OF THE ROOTS OF A QUADRATIC EQUATION 25

Ch. 4.4 cont HW: p. 254 # 1 – 2 & p. 215 # 13 26

CH. 3 & 4 REVIEW 27



3.2–PropertiesofQuadraticFunctions Recap: Function: A relation where there’s only ONE output for every valid input. Eg: y = 3x + 1 y = 2x2 Definitions: A quadratic function is any function that can be written in:

Standard Form: cbxaxy 2

Vertex Form: qpxay 2)(

Or any other forms that can be transformed into either Standard Form. Where

cba and,, are constants

0a meaning 2x - term has to be present in the equation.

Identify Quadratic Functions: Quadratic Functions? (y = ax2 + bx + c) yes/no 1

2

3

4

The Graph of Quadratic Functions:

The graph of every quadratic function is a curve called a ______________________.

The ______________ of a parabola is its highest or lowest point. The vertex may be a ________________ point or a ________________ point.

The axis of symmetry intersects the parabola at the vertex. The parabola is symmetrical about this line.



Graph of Quadratic Functions Identify the following:

1) 2xy

Equation of an Axis of Symmetry Coordinates of the Vertex: x- intercept: y-intercept:

Domain:

Range:

2) y = x2 + 6x + 5


Domain:

Range:



3) y = 3

1 (x +2)(x - 4)


Domain:

Range:

Using Technology to graph Use technology to sketch each function. Identify the following.

1) 4)1(2 2 xy


Domain:

Range:

Ch. 3. 2 HW: p. 174 # 1 - 3, 5a,b, 6a, 7, 8



3.1–InvestigatingQuadraticFunctionsinVertexForm Goal of this lesson: After this lesson, you should be able to graph any quadratic functions of the form

qpxay 2)( without using a table of values. y = x2 Table of Values

x y -3 -2 -1 0 1 2 3

Graph of y = x2

Movement of the parabola, y = x2

y = ax2 (|a| > 1) Horizontal compression or vertical stretch When a is positive and bigger than 1: y = 2x2 Using a table of values, graph y = 2x2

When a is negative and smaller than -1: y = -2x2 Using a table of values, graph y = -2x2

Is the graph wider/narrower than y = x2? Is the graph facing up/down?




y = ax2 (0 < |a| < 1) Horizontal stretch or vertical compression When a > 0 (a is positive)

y = 2

1x2

Using a table of values, graph y = 2

1x2

When a < 0 (a is negative)

y = 2

1 x2

Using a table of values, graph y = 2

1 x2



More Practice: Graph y = 3x2 Predict: Horizontal stretch or compression? Is the graph facing up/down?

More Practice: Graph y = 3

1 x2

Predict: Horizontal stretch or compression? Is the graph facing up/down?



More Practice: Graph y = 3

4x2

Predict: Horizontal stretch or compression? Is the graph facing up/down?

More Practice: Graph y = 2.0 x2 Predict: Horizontal stretch or compression? Is the graph facing up/down?

y = (x – p)2 Horizontal Shift When p is negative: y = (x + 2)2 Using a table of values, graph y = (x + 2)2

When p is positive y = (x - 2)2 Using a table of values, graph y = (x - 2)2

The graph of 2xy shifts horizontally to the ___________ when p is negative.

The graph of 2xy shifts horizontally to the ___________ when p is positive.



y = x2 + q Vertical Shift When q is positive: y = x2 + 2 Using a table of values, graph y = x2 + 2

When p is negative y = x2 - 2 Using a table of values, graph y = x2 - 2

The graph of 2xy shifts vertically ___________ when q is positive.

The graph of 2xy shifts vertically ___________ when q is negative.

Putting Everything Together: y = a(x – p)2 + q

Steps 1) Horizontal Shift 2) Vertical Shift Answer 1 and 2 to find the vertex: Vertex: ( , ) 3) Horizontal Stretch/Compression? 4) Facing Up or Facing Down?

Graph y = 2(x+3)2 + 1



Putting Everything Together: y = a(x – p)2 + q Steps

1) Horizontal Shift 2) Vertical Shift Answer 1 and 2 to find the vertex: Vertex: ( , ) 3) Horizontal Stretch/Compression? 4) Facing Up or Facing Down?

Graph

y = 2

1 (x - 1)2 + 3

Steps

1) Horizontal Shift 2) Vertical Shift Answer 1 and 2 to find the vertex: Vertex: ( , ) 3) Horizontal Stretch/Compression? 4) Facing Up or Facing Down?

Graph y = 25.0 (x + 2)2 – 4

Ch. 3.1 HW: p. 157 #1 – 9 odd letters



3.3–CompletingtheSquare Recap - Algebra Tiles:

x2 -x2 x -x 1 -1 Recap - Concept of Zero Pair: + = 1 + (-1) = 0 + = x + (-x) = 0 + = x2 + (-x2) = 0 Recap - Modeling Polynomials Using Algebra Tiles: 1) Model each polynomial using algebra tiles.

a) 122 xx

Can you form a square using these tiles?

b) 962 xx

Can you form a square using these tiles?

2) Write the polynomial expression for the following:

a)

b)



Graphing quadratic equations of the form cbxaxy 2 can be time consuming. However, you

have learnt how to graph quadratic equations of the form qpxay 2)( quickly. The goal of this lesson is to:

- Transform cbxaxy 2 into qpxay 2)(

- Identify vertex and be able to sketch the graph quickly. Completing the Square: Modeling with Algebra Tiles Algebra Tiles Method Algebra Method

Model the polynomial, 562 xx using algebra tiles.

Can you form a square using these tiles? If no, what do you need to do?



Completing the Squares: Algebraic Steps

1) Convert 22122 2 xxy to vertex form.

2) Convert 23284 2 xxy to vertex form.

3) Determine the coordinates of the vertex of 322

1 2 xxy . What is the maximum or minimum

value?

4) Convert xxy8

32 to vertex form. Is the graph facing up or down? What is the vertex? What

is the max. or min. value?



Word Problems: 1) The sum of two numbers is 40. Their product is a maximum. Determine the numbers that produce

the maximum product. What is the maximum product? 2) 120 meters of fencing are available to enclose a rectangular area.

a) What is the maximum area that can be enclosed? b) What dimensions produce the maximum area? c) State the domain and range for this problem.



3) When bicycles are sold for $300 each, a cycle store can sell 200 in a season. For every $20 increase in the price, the number sold drops by 10. a) What price will provide the maximum revenue? b) What is the maximum revenue?

Ch. 3.3 HW: p. 192 #1 – 4, 6 – 8 odd letters



4.1–GraphicalSolutionsofQuadraticEquations Recap: Quadratic Function is a second degree polynomial function that can be written in the form:

cbxaxy 2 where a ,b and c are constants, and 0a . Quadratic Equations: Quadratic Equation is a second-degree equation that can be written in the standard form:

02 cbxax where a ,b and c are constants, and 0a . Ex:

0353 2 xx

Root of an equation: The solution(s) to an equation. Zero(s) of a function: The value(s) of x for which 0)( xf Identify x-intercept(s), root(s) of the equation, and zero(s) of the function: Quadratic Equations With One Real Root:

Given )(xf = 442 xx

x- intercept(s): zero(s) of the function, )(xf

root(s) of the equation, 0442 xx

Quadratic Equations With Two Different Real Roots:

Given )3)(1()( xxxf

x- intercept(s): zero(s) of the function, )(xf root(s) of the equation, 0)3)(1( xx

4)12)(3( xx



Quadratic Equations With No Real Root:

Given 42)( 2 xxxg

x- intercept(s): zero(s) of the function, )(xg

root(s) of the equation, 0422 xx

Solve each equation by graphing.

7422 xx Method One:

Graph 422 xxy Graph 7y and find the intersection points (find where the two graphs intersect).

Method Two:

Rearrange the equation 7422 xx to and solve for the x-intercepts.

Estimate the roots to the nearest tenth. You may use a graphing calculator.

4) What are the roots of the equation, 01682 xx ? How many different roots are there?

5) What are the zero(s) of the function, 1682 xxy ?

6) Solve 352 xx by graphing. Round to 4 decimals.



Word Problems: 7) The manager of Jasmine’s Fine Fashions is investigating the effect that raising or lowering dress

prices has on the daily revenue from dress sales. The function 215100)( xxxR gives the

store’s revenue R, in dollars, from dress sales, where x is the price change, in dollars.

a) What price changes will result in no revenue?

b) What price change will maximize the revenue?

Ch. 4.1 HW: p. 215 #1, #3a,b,e, 4 - 7, 10, 12



4.2–FactoringQuadraticEquationsRecap: Solve 1032 xx =0 Solve by graphing How do we solve the equation without a graphing device?

In order to solve quadratic equations by factoring, you need to recap how to factor various types of polynomials (especially, polynomials of degree 2). Examples: Factor Polynomials 1. Factor completely.

a) 1222 2 xx

b) Factor 34

1 2 xx

c) Factor 94 2 x

d) Factor 4416 yx

e) Factor 384 2 tt



Examples: Factor Polynomials of Quadratic Form

2. Factor 9)2(24)2(12 2 xx

3. Factor 22 )12(4)13(9 yx

Examples: Solving Quadratic Equations by Factoring

4. Solve 0523 2 xx

5. Solve 0592 2 xx

6. Solve 2)1(24)1)(2(3 xxx



7. Solve -0.01x2 + 0.84x = 0 Examples: Apply Quadratic Equations 8. An osprey, a fish-eating bird of prey, dives toward the water to catch a salmon. The height, h , in

metres, of the osprey above the water t seconds after it begins its dive can be approximated by the

function 45305)( 2 ttth . Determine the time it takes for the osprey to reach a height of 20

m.

9. An 18-m tall tree is broken during a severe storm, as shown. The distance from the base of the

trunk to the point where the tip touches the ground is 12 m. At what height did the tree break?

Ch. 4.2 HW: p. 229 #1 – 10 odd letters, 11, 14, 16, 17, 19a, 20, 27 odd letters



4.3–SolvingQuadraticEquationsbyCompletingtheSquare We’ve learnt how to solve quadratic equations by:

1) Graphing 2) Factoring

Today, we’ll learn how to solve quadratic equations by

3) Completing the square Recap: Solve

92 x Similarly: Solve

0162 x

025)2( 2 x Examples: Solve by Completing the Square

1) 021102 xx

2) 0732 2 xx



3) Brian is placing a photograph behind a 12-in by 12-in piece of matting. He positions the photo so the matting is twice as wide at the top and bottom as it is at the sides. The visible area of the photograph is 54 sq. in. What are the dimensions of the photograph?

Ch. 4.3 HW: p. 240 #2, 4, 5, 6 odd letters



4.4–QuadraticFormula Recap: Solve by Completing the Square Deriving the Formula

0172 2 xx

02 cbxax



Examples: Use the quadratic formula to solve each quadratic equation. Express your answer as exact and simplified roots.

1) 023 2 xx

2) 0762 xx

3) 0544 2 xx

Ch. 4.4 HW: p. 254 #3 – 5 odd letters, 13, 14, 18



4.4–TheNatureoftheRootsofaQuadraticEquation Recall: Quadratic equations can have: 2 distinct

roots 0 = x2 + x – 2

Use the quadratic formula to solve for x .

1 root y = x2 - 6x + 9


No real roots y = x2 -2x + 5


Discriminant: You can determine the nature of the roots for a quadratic equation by the value of the disciminant, b2 – 4ac:

When the discriminant is positive ____________, there are _________________ real roots



When the discriminant is zero ____________, there is ____________________ real root

When the discriminant is negative ____________, there are _________________ real roots. Examples: 1) Determine the nature of the roots (no real roots, 1 real roots, two different real roots).

a) x2 – 4x + 2 = 0 b) 2x2 – 5x + 8 = 0

c) x2 – 4x + 4 = 0

2) Given x2 + kx + 2 = 0

a) For what values of k does the equation have 1 real root? b) For what values of k does the equation have 2 different roots?

Ch. 4.4 cont HW: p. 254 # 1 – 2 & p. 215 # 13



Ch.3&4Review Multiple Choice Questions: 1. Which of the following is a quadratic function?

a) 23 xxy b) 130)( 2 xxxf

c) 12 xxy d) 13 xy

2. What is the vertex? Is the vertex a minimum or maximum point?

a) (2, 3) is a minimum point b) (-2, 3) is a maximum point

c) (3, 2) is a minimum point d) (-3, 2) is a maximum point

3. What is the axis of symmetry of 5)2(5.0)( 2 xxf

a) 5y b) 5y

c) 2x d) 2x

4. Which of the quadratic functions is congruent to 22xy ?

a) 12

1 2 xy b) 2)1(2 xy

c) 5)1()( 2 xxf d) 2)1(2

1 2 xy

5. What is the equation of the parabola with vertex (1, -3)?

a) 3)1( 2 xy b) 3)1( 2 xy

c) 1)3(2

1 2 xy d) 3)1(2

1 2 xy

6. Which quadratic function has the vertex at (2, -1) and passes through (3, 1)?

a) 1)2( 2 xy b) 1)2( 2 xy

c) 1)2(2 2 xy d) 2)1(2

1 2 xy



7. Which of the following is an equation of the quadratic function that has a maximum value of -2 at 3x ?

a) 2)3(2

1 2 xy b) 2)3( 2 xy

c) 2)3( 2 xy d) 2)3( 2 xy

8. Which graph is represented by y = 1)3(2 2 x

a)

b)

c)

d)

9. What is the quadratic function in vertex form for the parabola shown below?

a) 2)1(2 2 xy b) 2)1(2

1 2 xy

c) 2)1(4 2 xy d) 2)1(4

1 2 xy

10. What are the domain and range of 1)1(2 2 xy

a) Domain: }|{ xx

Range: }1|{ yy

b) Domain: }1|{ xx

Range: }1|{ yy

c) Domain: }1|{ xx

Range: }1|{ yy

d) Domain: }|{ xx

Range: }1|{ yy

11. The parabola 2xy is transformed as described below. Its image has the form qpxay 2)( .

Determine each value of a, p, and q for the following transformation: Shift 2 units left, 3 units down. Compress vertically by a factor of 4.

a) 4

1a , 2p , 3q

b) 4a , 2p , 3q

b) 4

1a , 2p , 3q c)

4

1a , 2p , 3q



12. What is the nature the roots of the quadratic equation: 0132 2 xx

a) There are two different roots b) There is one root c) There are no roots d) There are three different roots

Written Response (Show clear and concise work): 2 marks each 13. Sketch the function.

a) 2)3(5.0 2 xy b) 7)1(3

5 2 xy

Is this a vertical stretch/compression?

Is this a vertical stretch/compression?

14. Factor 14)1(11)1(2 2 xx .

15. Given the graph below, write an equation of the function in: a) y = a(x – p)2 + q form

b) y = ax2 + bx + c form



16. Determine the equation of a parabola (in y = a(x – p)2+ q form) with vertex (-1, 3) and passing through (0, 2).

17. Given the quadratic function, 2122 2 xxy

a) What are the coordinates of the vertex?

b) What is the maximum/minimum value?

18. Find the zeros of the function, 372 2 xxy

19. Find the roots of the equation, 0133 2 xx

20. A store sells energy bars for $2.25. At this price, the store sold an average of 120 bars per month

last year. The manager has been told that for every 5 ¢ decrease in price, he can expect the store to

sell eight more bars monthly.

Let = x be the number of times you decrease the price by 5 ¢.

Let = )(xR be the revenue as a function of x .

a) What quadratic function can you use to model revenue as a function of x ?

b) Use an algebraic method to determine the maximum revenue the manager can expect the store to achieve.

c) What price will give that maximum?



21. The parks department is planning a new flower bed. It will be rectangular with dimensions 9m by 6m. The flower bed will be surrounded by a grass strip of constant width with the same area as the flower bed. a) Write a quadratic equation to model the situation.

b) Solve the quadratic equation. Justify your choice of method.

c) Calculate the perimeter of the outside of the path.

Ch. 3 & 4 - Quadratic Functions and Equations Notes Block D · Chapter 3 & 4 – Quadratic...

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