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A. Basic Skills

1. Given the quadratic equation, y = -2x2 + 20x - 40, be able to do all of the following:

a) Put into vertex form using the pand qformulas.

b) Put into vertex form by completing the square.

c) Find the x-intercepts (zeroes) using algebra.

d) Find the y-intercept using algebra.

e) Draw a rough sketch of the parabola, using the vertex and intercepts.

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f) Graph the parabola with your calculator. Find the vertex and x-intercepts using your calculator.

g) Identify the domain and range of the parabola.

h) Identify the equation of the axis of symmetry.

B. Applications

Situations modelled by a quadratic.

Ex 1: The path of the ball for many golf shots can be modeled by a quadratic function. The path of agolf ball hit at an angle of about 10 to the horizontal can be modeled by the function

h(d) = - 0.002d2 + 0.4d

where h(d) is the height of the ball, in meters, and dis the horizontal distance the ball travels, in me-

ters, until it first hits the ground.

a) What is the maximum height reached by the ball?

b) What is the horizontal distance of the ball from the golfer when the ball reaches its maximumheight?

c) What distance does the ball travel horizontally until it first hits the ground?

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Max/Min questions

Ex. 2: A cattle farmer wants to build a rectangular fenced enclosure divided into five rectangularpens. A total length of 120 m of fencing material is available. Find the overall dimensions ofthe enclosure that will make the total area a maximum.

Ex. 3: The Environmental Club sells sweatshirts as a fund-raiser. They sell 1200 shirts a year at$20 each. They are planning to increase the price. A survey indicates that, for every $2 in-crease in price, there will be a drop of 60 sales a year. What should the selling price be in

order to maximize the revenue?

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A. The Basics

1. Given the quadratic equation, y = -2x2 + 20x - 40, be able to do all of the following:

a) Put into vertex form using the pand qformulas.

b) Put into vertex form by completing the square.

c) Find the x-intercepts (zeroes) using algebra.

d) Find the y-intercept using algebra.

e) Draw a rough sketch of the parabola, using the vertex and inter-cepts.

2

b

p a

=

2

q c ap=

2 0

2 ( 2 )

5

p

p

=

=

24 0 ( 2 )(5)

4 0 ( 50)

10

q

q

q

=

=

=

22( 5) 10y x= +

2

2

2

2

2

2 20 40

2( 10 25 25) 40

2( 10 25) 2( 25) 4 0

2( 10 25) 50 40

2( 5) 10

y x x

y x x

y x x

y x x

y x

= +

= +

= +

= + +

= +

2

2

2

2

2( 5) 1 0

1 0 2( 5)

1 0( 5)

2

5 ( 5)

y x

x

x

x

= +

=

=

=

25 ( 5)x =

5 5

2.7639

x

x

=

=

5 5

7.2361

x

x

=

=

22(0) 20(0) 40

40

y

y

= +

=

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f) Graph the parabola with your calculator. Find the vertex and x-intercepts using your calculator.

g) Identify the domain and range of the parabola.

h) Identify the equation of the axis of symmetry.

B. Applications

Put into vertex form, find the zeroes, and sketch graph.

a) What is the maximum height reached by the ball?

Max height is 20 m.

b) What is the horizontal distance of the ball from thegolfer when the ball reaches its maximum height?

The ball will be 100 m away.

c) What distance does the ball travel horizontally until it first hits the ground?

The ball travels 200 m horizontally.

:

: { / }: { / , 10}

Set Notation

D x x

R y y y

:

: ( , ): ( ,10]

Interval Notation

D

R

5x=

0.4

2( 0.002)

100

p

p

=

=

( ) 2h d 0 .0 0 2 d 0 .4 d 0= + +

20 ( 0.002)(100)

20

q

q

=

=

( ) 2h d 0.002(d - 100) 20= +

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Max/Min questions

Ex. 2: A cattle farmer wants to build a rectangular fenced enclosure divided into five rectangularpens. A total length of 120 m of fencing material is available. Find the overall dimensions ofthe enclosure that will make the total area a maximum.

Let x = the length of a pen.

Since there is 120 m of fence to work with, thereis (120 - 6x) metres left over to split amongst thetop and bottom...

The total area of the five pens will be: A = (x)(60 - 3x)

Simplify and put into vertex form:

Since the parabola opens, down, the vertex (10, 300) is a maximum. The length of a pen is

10 m. The maximum area is 300 m2.

If we substitute x = 10 into 60 - 3x, we get 30 m. The dimensions of the pen are 10 m x 30 m.

Ex. 3: The Environmental Club sells sweatshirts as a fund-raiser. They sell 1200 shirts a year at$20 each. They are planning to increase the price. A survey indicates that, for every $2 in-crease in price, there will be a drop of 60 sales a year. What should the selling price be inorder to maximize the revenue?

x = # of increases

price of a shirt = $(20 + 2x)

number of shirts that will sell = (1200 - 60x)

Maximum occurs at the vertex (5, 27000).

The price, if increased 5 times, will be $30.

The maximum revenue will be $27000

x x x x x x

120 6

2

60 3

x

x

=

60 - 3x

60 - 3x

2

2

60 3

3 60

A x x

A x x

=

= +

2

2

2

2

2

3 60

3( 20 )

3( 20 100 100)

3( 20 100) 300

3( 10) 300

A x x

A x x

A x x

A x x

A x

= +

=

= +

= + +

= +

2

2

2

2

2

Re (20 2 )(1200 60 )

24000 1200 2400 120

120 1200 24000

120( 10 25 25) 24000

120( 10 25) 120( 25) 24000120( 5) 27000

Total venue x x

R x x x

R x x

R x x

R x xR x

= +

= +

= + +

= + +

= + += +