UNIT 4 WORKING WITH QUADRATIC MODELS Date Lesson …mrkennedy.pbworks.com/w/file/fetch/116576283/MCF 3M Unit 4 Shell.pdfMCF 3M Lesson 4.0 Getting Started Ex. 1 Match the term with

UNIT 4 – WORKING WITH QUADRATIC MODELS

Date Lesson Text TOPIC Homework

Mar.

27 4.0 Opt

Getting Started Pg. 192 # 1 – 7

Mar.

28

4.1

4.1

The Vertex Form of a Quadratic Function Pg. 203 # 1 – 4, 6 – 12, 14

Mar.

29

4.2

4.2

Relating the Standard & Vertex Forms:

Completing the Square (CTS)

Pg. 214 # 2 - 5

Mar.

30

4.2 (II)

4.2

Relating the Standard & Vertex Forms:

Completing the Square

Pg. 214 # 6 - 8, 10 - 13

Mar.

31

4.3

4.3

Solving Quadratic Equations using the

Quadratic Formula

CTS QUIZ

Pg. 222 # 1 – 3, 5, 6, 8 - 11

Apr. 3 4.4

Mid-Chapter Review

CTS QUIZ

Pg. 226 # 1 - 11

Apr. 4 4.5

4.4

Investigating the Nature of the Roots

QUIZ (4.1 – 4.3)

CTS QUIZ

Pg. 232 # 2 – 10, 13, 14

Apr. 5 4.6

4.5

Using Quadratic Function Models to Solve

Problems

Pg. 239 # 1 – 9, 11, 13

Apr. 6 4.7

4.6

Using the Vertex Form to Create Quadratic

Function Models from Data

Pg. 250 # 3 – 6, 8, 12

Apr. 7 4.8

Review for Unit 4 Test Pg. 254 # 1 - 11

Apr.

11

4.9

(39) UNIT 4 TEST

MCF 3M Lesson 4.0 Getting Started

Ex. 1 Match the term with the picture or example that best illustrates its definition.

Ex. 2 Solve 01522 xx by factoring.

Ex. 3 Factor the following perfect square trinomials.

a) 25102 xx b) 92416 2 xx

Pg. 192 # 1 – 7

MCF 3M Lesson 4.1 The Vertex Form of a Quadratic Function

Vertex Form of a Quadratic Function - is in the form khxay 2)( , where the vertex is (h, k)

- the equation of the Axis of Symmetry is x = h

- if a > 0 the parabola opens up & the minimum value is y = k

- if a < 0 the parabola opens down & the maximum value is y = k

Ex. 1 Mr. Call wants to surround his rose garden with 80 patio stones that are each 1 m square. He wants his

rose garden to have the largest possible area.

Two of his students determine quadratic functions to model the area of the rose garden.

Veena: wwwf 40)( 2 Rhoda: 400)20()( 2 wwg

w represents the width of the garden in metres and f(w) and g(w) both represent the area in m2.

Which function should the environment club use?

Ex. 2 What is the maximum area of a garden defined by wwwf 40)( 2 . How does this relate to the

function 400)20()( 2 wwg ?

Ex. 3 Determine the direction of opening, the equation of the axis of symmetry, the minimum value, the

vertex, the domain and the range of the quadratic function 6)4(2)( 2 xxf . Graph the function.

1 2 3 4 5 6 7 8 9–1–2–3–4–5 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

Ex. 4 Determine the zeros of the function 25)2()( 2 xxf .

THERE ARE TWO METHODS WE COULD USE:

Expand and Factor

Rearrange to solve for x.

Ex. 5 Determine the equation in vertex form of the quadratic function shown.

1 2 3 4 5 6 7–1–2–3–4–5–6–7 x

1

2

3

4

5

6

7

–1

–2

–3

–4

–5

–6

–7

y

Pg. 203 # 1 – 4, 6 – 12, 14

MCF 3M Lesson 4.2 Standard to Vertex Form: Completing the Square

Completing the Square - the process of adding a constant to a given quadratic expression to

form a perfect trinomial square

for example, x2 + 6x + 2 is not a perfect square, but if 7 is added to

it, it becomes x2 + 6x + 9, which is (x + 3)

2

Ex. 1 Find the vertex form of each of the following by completing the square.

a) 36)( 2 xxxf

b) 4122 2 xxy

c) 432)( 2 xxxf d) 542

1)( 2 xxxf

Ex. 2 Judy wants to fence three sides of the yard in front of her house. She bought 60 m of fence and wants

the maximum area she can fence in. The quadratic function 2260)( xxxf ,where x is the width of the

yard in metres, represents the area to be enclosed. Write an equation in vertex form that gives the

maximum area that can be enclosed.

(Day 1) Pg. 214 # 2 – 5

(Day 2) Pg. 214 # 6 – 8, 10 – 13

MCF 3M Lesson 4.2 Standard to Vertex Form: Completing the Square (Part II)

Ex. 1 Find the vertex form of each of the following by completing the square.

a) 3162 2 xxy b) 132 2 xxy

c) 243

1 2 xxy d) 132

1 2 xxy

Pg. 214 # 6, 7, 8, 10, 11, 12, 13

MCF 3M Lesson 4.3 The Quadratic Formula

Ex. 1 A quarter is thrown form a bridge 14 m above a river. The height of the quarter h(t), in metres above

the water, at time t, in seconds, is given by the quadratic function 14105)( 2 ttth . When does the

quarter hit the water?

h(t) = 0 when it hits the water

Can we factor 141050 2 tt ?

We would not want to have to do this every time we had a quadratic function that could not be factored. There has to be a quicker way.

Quadratic Formula

Ex. 2 A quarter is thrown form a bridge 14 m above a river. The height of the quarter h(t), in metres above

the water, at time t, in seconds, is given by the quadratic function 14105)( 2 ttth .

Use the quadratic formula to determine when the quarter hits the water?

Ex. 3 Use the quadratic formula to solve each of the following equations.

State your answer correct to 2 decimal places when necessary.

a) 0225302 xx b) 01523 2 xx

c) 152 2 xx

Ex. 4 The profit from the production of the play this year is modelled by the quadratic equation

100079060)( 2 xxxP , where P(x) is the profit in dollars and x is the price of a ticket in dollars.

a) Use the quadratic formula to determine the break-even price for the tickets.

b) What ticket price should be charged to maximize the profit?

Pg. 222 # 1 – 3, 5, 6, 8 - 11

MCF 3M Lesson 4.4 Mid-Chapter Review

Pg. 226 # 1 - 11

MCF 3M Lesson 4.5 Nature of the Roots of a Quadratic Function

Ex. 1 Use the QUADRATIC FORMULA to determine the number of roots (zeros, x-intercepts) for each of the

following quadratic equations.

a) 2282 xxy

b) 962 xxy

c) 635 xxy

Generally, a quick way to determine the number of zeros of a quadratic function is to examine the

DISCRIMINANT.

DISCRIMINANT (b2 – 4ac)

Number of Real

Zeros (Roots)

a)

b)

c)

Ex. 2 Use the discriminant to determine the number of zeros of:

a) 4429 2 xxy b) 1053 2 xx c) 3)2()( 2 xxf

Ex. 3 For what value of k will 0652 xkx have no zeros?

Ex. An arrow is released with an initial speed of 39.2 m/s. It travels according to 3.12.399.4 2 ttth ,

where h is the height reached, in metres, and t is the time taken, in seconds. Will the arrow ever reach a

height of 80 metres?

Pg. 232 # 2 – 10, 13, 14

MCF 3M Lesson 4.6 Using Quadratic Models to Solve Problems

Ex. 1 The graph shows the height of a rock launched from a slingshot, where time, t, is in seconds and height,

h(t), is in metres. Determine when the rock hits the ground. State your answer correct to 1 decimal place.

Ex. 2 Mr. Gala has an apple orchard with 90 trees. He earns an annual revenue of $120 per tree. If he plants

more trees, they have less room to grow and produce fewer apples. As a result he expects the annual

revenue per tree to be reduced by $1 for each additional tree. The revenue function for his orchard is

modelled by the function )120)(90()( xxxR , where x is the number of additional trees planted.

Regardless of the number of trees planted, the cost of maintaining each tree is $8. How many trees

must Mr. Gala plant to maximize profit for his orchard?

N.B: Profit = Revenue - Cost

Ex. 3 The cost of running a Teddy Bear production line is modelled by the function 212.128.0)( 2 xxxC ,

where C(x) is the cost per hour in thousands of dollars, and x is the number of Teddy Bears produced per

hour in thousands. Determine the most economical production level.

Ex. 4 A company that makes widgets uses the relation 200)11(2)( 2 xxP , where P(x) is yearly profit in

thousands of dollars, and x is numbers of widgets sold, in thousands. How many widgets must the company

produce in order to have a profit of $150 000.

Pg. 239 # 1 – 9, 11, 13

MCF 3M Lesson 4.7 Using Vertex Form to Create Models from Data

Ex. 1 Determine a quadratic equation in vertex form that best represents the arch between the towers in the

suspension bridge photo. Express your equation in standard form as well.

Picture of Bridge Picture of Bridge with Grid

Ex. 2 A hose sprays a stream of water across a lawn. The table shows the approximate height of the stream

above the lawn at various distances from the person holding the nozzle. Write an algebraic model in

vertex form that relates the height of the water to the distance from the person.

Check your answer using a graphing calculator.

Make a scatter plot. Determine the equation of the curve of best fit.

Draw a curve of best fit.

1 2 3 4 5 6 7 8 9 10 x

1

2

3

4y

b) Using your equation, determine,

to the nearest 10th of a second,

when the height of the water is 2 m.

Pg. 250 # 3 – 6, 8, 12