UNIT 3 – QUADRATIC FUNCTIONS

Date Lesson Text TOPIC Homework

Mar. 6 3.1

(21) 3.1

Properties of Quadratic Functions STUDY FOR TEST # 2

Mar. 8 3.2

(22) 3.2

Maximum and Minimums of Quadratic

Functions

Pg. 145 # 2 – 5, 7, 8

Pg . Pg. 153 # 1 – 4

P Pg. 170 # 5abc

Mar.

18

3.3

(23) 3.2

Maximum and Minimums of Quadratic

Functions (Word Problems)

Q4(Complete the Square)

Pg. 154 # 8 – 11

Pg. 170 # 6, 7

Pg. 147 # 11, 12

Mar.

19

3.4

(24) 3.4

Radicals

Q4(Complete the Square)

Pg. 167 # (1 – 7)sodo, 11, 12, 14

WS 3.4 # 8 - 11

Mar.

20

3.5

(25) 3.5

Solving Quadratic Equations

Q4(Complete the Square)

Pg. 177 # 1 – 5, 7, 8

Mar.

21

3.6

(26) 3.5

Solving Quadratic Equations

Q5(3.1 – 3.4)

WS 3.6 # 5, 15, 28, 12acde

Mar.

22

3.7

(28) 3.5

Quadratic Word Problems Pg, 178 # 9 – 13, 16, 17

Mar.

25

3.8

(29) 3.5

Quadratic Word Problems II WS 3.8

Mar.

26

3.9

(30) 3.6

Zeros of a Quadratic Function

Q6(3.5 – 3.8)

Pg. 185 # 3, 4, 6 – 11, 14 - 16

Mar.

27

3.10

(31) 3.7

Finding the Equation of a Quadratic

Relation

Pg. 192 # 2 – 9, 16, 17

Mar.

28

3.11

(32) 3.8

Linear – Quadratic Systems Pg. 198 # 4, 10, 12, 14

Mar.

29

3.12

(33) 3.3

Inverse of a Quadratic Function

Q7(3.9 – 3.10)

Pg. 160 # 2 – 8, 10, 17

Apr. 1 3.14

(34)

Review for Unit 3 Test Pg. 202 # 4, 5, 9, 11 – 15,

18 – 22, 23a

Apr. 3 3.15

(35)

UNIT 3 TEST

MCR 3U Lesson 3.1 Properties of Quadratic Functions

Function: 432 xy 132 xxy

Graph:

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

2 4 6 8–2–4–6–8 x

2

4

6

8

–2

–4

–6

–8

y

Vertex:

Zeroes:

Equation

of Axis of

Symmetry

Domain

Range

Properties of Quadratic Functions:

have no restrictions on the domain

constant non-zero 2nd differences (2)

have a degree of 2

open up if a > 0 or 2 > 0

open down if a < 0 or 2 < 0

have different algebraic forms

o standard form 0,2 acbxaxy

o factored form 0),)(( atxsxay

o vertex form 0,)( 2 akhxay

Symmetrical points are points which share the same y value and are on opposite sides of the axis

of symmetry. ie: the points (3, 9) and (-3, 9) are symmetrical points for the parabola y = x2

NEW METHOD OF FINDING VERTEX

Partial Factoring: common factor the bxax 2 terms to

determine 2 symmetrically opposite x

values (ie: same y value), then averaging

them to find axis of symmetry gives x

coordinate of vertex

Complete the table below:

Function Opens: Symmetrical

Points Vertex

Axis of

Symmetry Domain Range

Max/Min

Point

VERTEX FORM

a) 5)1(2 2 xy

b) 2)1(3 xy

FACTORED FORM

c) )5)(1(2 xxy

d) )5)(2( xxy

PARTIALLY FACTORED

FORM

e) 8)2( xxy

f) 9)4(2 xxy

STANDARD FORM

g) 2142 xxy

h) 2242 xxy

Ex. For each of the following quadratic functions, state the vertex and whether it is a maximum or minimum.

a) 6)2(2 xxy b) 1123 2 xxy

MCR3U Lesson 3.2 Maximum/ Minimum Values of a Quadratic Function

Determine the maximum or minimum value of each quadratic function using the indicated method.

(i) Completing the Square

a) 782 xxy b) 11122 2 xxy

(ii) Partial Factoring

a) 822 xxy b) 5124 2 xxy

HARDER ONES

Ex. Find the vertex of each of the following by completing the square. NO CALCULATORS.

a) 152

1 2 xxy

b) 732 2 xxy

c) 46

5

3

2 2 xxy

HW: Pg. 145 # 2 – 5, 7abcd, 8abcde

Pg. 153 # 1 – 4

Pg. 170 # 5abc

The Messy ones! WS 3.3

MCR3U Lesson 3.3 Maximum/ Minimum Values of a Quadratic Function

Determine the maximum or minimum value of each quadratic function using the indicated method.

(iii) partial factoring (iv) completing the square

a) 682 2 xxy b) 1253)( 2 xxxf

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 height in metres and t is time in seconds. What is the maximum height the arrow will reach and

after how long does this occur?

Pg. 154 # 8 – 11; Pg. 147 # 11, 12

Pg. 170 # 6, 7

WS 3.3 - practice hard completing the square

MCR 3U Lesson 3.4 Radicals

Radical – a square, cube, or higher root, such as 24 , or 3273 , or 56254 .

RADICAL

index

n a radicand

Ex. 1 Convert each radical from entire to mixed:

a) 50 b) 45 c) 128 d) 108

Adding and Subtracting Radicals

Generally: To add/subtract radicals the radicals must have the same index and the same radicand.

The coefficients are added/subtracted and the radicands remain the same.

Ex. Simplify

a) 56255322 b) 50328

c) 2

321222785

Multiplying Radicals

Generally: If the radicals have the same index, products are found by multiplying the radicands.

Ex. Simplify:

a) 5432 b) 32283 c) 234

d) 3432 e) 2525

f) 531025232

g) 20 10

5 2 h)

40

8

Rationalizing Denominators – A radical is not in simplest form if there is a radical in the denominator.

To eliminate this, we can multiply the radical by a form of 1 that will eliminate

the radical in the denominator. When the denominator is a binomial, we multiply by

what is called the conjugate ( same terms with the opposite operation).

The product of any conjugate pair is always a difference of squares.

Ex. Simplify each of the following, writing your answer with a positive denominator:

a) 3

1 b

5

34 c)

32

23

d) 52

523

e)

3524

32

Pg. 167 # (1 – 7)sodo, 11, 12, 14

WS 3.4 # 8 - 11

MCR3U Lesson 3.5 Solving Quadratic Equations

Ex. 1 Find the x-intercepts of each of the following:

a) 0322 xx b) 01572 2 xx

c) 7113 2 xx

Ex. 2 Solve each of the following.

a) 242)3(4 nnn b) 0962 xx

Pg. 177 # 1 – 5, 7, 8

MCR3U Lesson 3.6 Solving Quadratic Equations (II)

Ex. Solve each of the following:

a) n

n

n

nn

4

32

3

1

2

1

b) 286

6

2

52

xxx

WS 3.6 # 5, 15, 28, 12acde

MCR 3U Lesson 3.7/3.8 Solving Quadratic Word Problems

Ex. 1 When two consecutive numbers are squared and the squares added, their sum is 421.

What are the possible numbers?

Ex. 2 A landscaper wishes to plant a boundary of tulips within a rectangular garden with dimensions

18 m by 12 m. To obtain a pleasing look the area of the tulip border should be half the area of

the garden. How wide should the border be, correct to 1 decimal place?

Ex. 3 The flight of a bowling ball is modelled by the equation 15105)( 2 ttth .

a) How long will the bowling ball be in the air? b) What is the maximum height of the ball?

c) What is the height of the ball when

released?

d) For how long, correct to 2 decimal places, will the bowling ball be at a height of at least 18 cm?

(3.7) Pg. 178 # 9 - 13, 16, 17

(3.8) WS 3.8

0.5 1 1.5 2 2.5 3 3.5 t

10

20

30

40

h( t)

MCR3U Lesson 3.9 Zeros of a Quadratic Function

Ex. Use the QUADRATIC FORMULA to determine the number of 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 ( acbD 42 ) Number of Real Zeros

or (Roots) or [x-intercepts]

a)

b)

c)

Ex. Use the discriminant to determine the number of solutions of:

a) 049429 2 xx b) 1053 2 xx

Ex. 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. 185 #3, 4, 6-11, 14 - 16

MCR3U Lesson 3.10 Finding The Equation of a Quadratic Relation

Ex. A parabolic bridge over a river has a span of 100 metres and a height of 30 metres. What is the

maximum height of a boat that can go under the bridge at a point 20 metres from the shoreline?

Ex. What characteristics will the parabolas 413)( xxxf and 412)( xxxg share?

Ex. Determine the equation of the parabola

a) that passes through )0,4( , )6,2( , and )0,7( . b) with vertex )3,2( and passing through )16,3( .

c) with x-intercepts 32 and 32 and passing through (4, 8).

Pg. 192 # 2 - 9, 16, 1 7

1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8–9 x

1

2

3

4

5

6

7

8

9

–1

–2

–3

–4

–5

–6

–7

–8

–9

y

MCR3U Lesson 3.11 Linear – Quadratic Systems

Ex. Solve each of the following both by GRAPHING and ALGEBRAICALLY:

1. 32

1

2

xxy

xy

2. 1

4

3

582 2

xy

xxy

Ex. Determine the number of points of intersection of the following linear-quadratic system.

82)(

14123 2

xxg

xxxf

Ex. Determine the equation of the line that has a slope of -6 and intersects the function 763 2 xxxf

once.

Pg. 198 # 4, 10, 12, 14

5 10–5–10 x

5

10

–5

–10

y

5 10–5–10 x

5

10

–5

–10

y

MCR 3U Lesson 3.12 Inverse of a Quadratic Function

Ex. For the function 22 xxf ,

a) graph f.

b) graph 1f .

c) Determine the defining equation of 1f

by interchanging the variables.

Ex. For the function 7122 xxg ,

a) Try to determine the defining equation of

1g by switching the operations.

b) Graph g and 1g to check.

Ex. How could we restrict the domain of 1232 xxh so that 1h would be a function?

5 10–5–10 x

5

10

–5

–10

y

Ex. Find the defining equation of 1h where 2,1232

xxxh

Ex. For the function 13 xxf ,

a) graph f.

b) graph 1f .

c) Determine the defining equation of 1f .

Pg. 160 #2 - 8, 10, 17