fxx  1()2 fxax () 

Quadratic Functions

1 - Investigating Quadratic Functions in Vertex Form (Part 1) Part A: Compare the Graphs of ^{f x( )x2} and ^{f x( )ax2} where a0

1. a) Graph the following functions on the same set of coordinate axes, using a table of values.

b) What happens to the shape of the graph when:

a > 1? 0 < a < 1? a < 0?

Part B: Compare the Graphs of ^{f x( )x2} and ^{f x( )x2q} x f(x)=

3

−2

−1

0 1

x f(x)=

3

−2

−1 0 1 x f(x)=

3

−2

−1 0 1 x f(x)=

3

−2

−1

0 1

Functions ( ) 2

f x x ( ) 2 2

f x  x 1 2

( ) 2

f x  x

( ) 2 2

f x   x

b) Describe how the graph changes when:

q > 0 q < 0

Part C: Compare the Graphs of ^{f x( )x2} and ^{f x( ) ( x p )2}

1. a) Graph the following 3 functions on the same set of coordinate axes, using a table of values.

b) Describe how the graph changes when:

x f(x)=

−2

−1

0 1

x f(x)=

−2

−1

0 1

x f(x)=

−2

−1 0 1

x f(x)=

−2

−1

0 1 x f(x)=

−2

−1

0 1

x f(x)=

−2

−1

0 1 Functions ( ) 2

f x x ( ) 2 4 f x x 

( ) 2 3 f x x 

Functions ( ) 2

f x x ( ) ( 2)2

f x  x ( ) ( 1)2

f x  x

Practice Exercises: Sketch the graph of each parabola without a calculator or table of values (just looking for a Pattern)

1.

( ) 2 2 f x x 

2.

3 2

y x

3.

2 4

y x 

4.

( ) 2 3 f x   x

5.

( 2)2

y x

6.

( ) ( 1)2

f x   x

Quadratic Functions

2 - Investigating Quadratic Functions in Vertex Form (Part 2) Definition: A quadratic Function is a function given by a polynomial of degree two.

f(x) = a(x - p)

+ q

Ex 1. Graph the following functions without using a table of values.

a) y = (x – 1)² + 4 b) y = - (x – 4)² - 1 c) y = 3(x + 3)² - 5 d) y = ½(x - 2)² + 4

Ex 2. Determine the following characteristics for each function.

Function Value

of a

Value of p

Value

of q Vertex Axis of Symmetry

Direction of

Opening Max/min y – int y=( x−2)²+4

y=−3( x+3)²+4

y=1

2(x−4 )²−3

Ex 3. What are the equations of the functions shown?

Quadratic Functions 3 - Word Problems

1) The following function gives the height h(t) in metres of a batted baseball as a function of time, t seconds, since the ball was hit: h(t)=−5(t−2)²+21 . Graph the function.

a) What was the maximum height of the ball?

b) What was the height of the ball when it was hit?

c) How long did it take to reach its maximum height?

d) Approximately how many seconds after it was hit did the ball hit the ground?

e) Find the height of the ball 1 second after it was hit.

2) The Hubble Space telescope is able to focus light from distant stars and galaxies. It does this, not with a lens like you would find in a magnifying glass, but with an enormous parabolic mirror as shown below.

The mirror is 2.4 m across and has a depth of 0.1 m. Draw a function that describes the parabolic mirror and determine an equation that describes it.

Parabolic Functions

4 - Investigating Quadratic Functions in Standard Form Key Concepts:

Ex. Write the equation y = (x - 2)² + 3 in expanded form

Quadratic Function in Standard Form:

f(x) = ax

+ bx + c

● a determines the ____________ and ___________ of the graph

● b determines the ________________ of the graph

● c determines the ____________________ of the graph

Converting the function from Vertex Form to Standard form Convert the following quadratic function into standard form:

a) f ( x )=−3 ( x +1)²−5 b) f ( x )=1

2( x−2)²+3

Converting the function from Standard form to Vertex form (Using a formula):

Development of the formula to find p and q:

Example: f(x) = – x² – 2x + 3

Practice:

Covert the following quadratic function from standard to vertex form using the formula:

a) f ( x )=3 x²−6 x +4 b) f ( x )=−1

2 x²+4 x + 2

Parabolic Functions 5 - Using Graphing Calculators

Graphing a function:

To graph a given function, follow these steps:

1. Enter the function into the Y= screen 2. Press Graph

3. You may need to adjust the viewing window to see the details of the graph appropriately!

Example: Graph the function

2

5 4 y   x   x

Adjusting the Viewing Window:

To adjust the viewing window, follow these steps:

1. Press WINDOW

2. Set each variable to the desired value (you may need to play around a bit to find what these desired values should be!)

Using the Calc menu to determine characteristics of the graph:

Another very useful feature is the CALC menu. You can use this menu to determine maximum or minimum points, zeros, intersection points and many other characteristics of a given graph. To do this follow the steps outlined below:

1. Enter the function into the Y= screen and press GRAPH.

2. Set the viewing window appropriately (so you can see the point of interest) 3. Press 2nd TRACE (CALC)

4. Select the menu item you are interested in (ie: for a zero, select 2: zero)

Ex 1. Graph the function defined by the equation

2

5 4

y x    x

Determine :

a) To find the x-intercept(s), use the zero feature from the CALC menu ________________

To find the y-intercept(s), use the value feature from the CALC menu _______________

b) To find the coordinates of the vertex,

use minimum/maximum feature from the CALC menu ___________________________

c) To find the equation of the axis of symmetry, use the x- coordinate of the vertex________

d) To find the domain and range use the information you’ve found ___________________

Ex 2. yx²2x5 vertex:

the y-intercept:

the x-intercepts:

Ex 3. y7x0.5x²3 vertex:

the y-intercept:

the x-intercepts:

x [ , ]y [ , ] ]

Y1 =

x [ , ]y [ , ] ]

Y1 =

Problem Solving

Solve using a graphing calculator. (Answers to two decimal places)

1. A rock is thrown off a cliff. The height, in metres, with respect to time, in seconds, is defined by the quadratic function: h(t) = -4.9t² + 12t - 16

a) What is the maximum height? _______________

b) When does it reach this height? _______________

c) How long does it take to reach the ground? _______________

d) How high is the cliff? _______________

e) What is the domain of this function? _______________

2. A rectangular pen is to be built along the side of a barn to house chickens.

a) Find the maximum area that can be enclosed with 60 m of fencing if the barn is one side of the enclosure.

b) What are the dimensions that give the maximum area?

x [ , ]y [ , ] ]

Y1 =

x [ , ]y [ , ] ]

Y1 =

Quadratic Functions 6 - Completing the Square We have seen that quadratic functions can be expressed in either:

standard form, f (x) = ax² + bx + c, or vertex form, f (x) = a(x – p)² + q.

Remember this?

a) (x + 3)

= b) (x – 3)

= c) (x + 13)

=

Now working backward.

(x _______)

= x

+ 18x __________

(x _______)

= x

- 14x __________

(x _______)

= x

__________ + 36 (x _______)

= x

__________ + 144

Ex: Rewrite each function in vertex form by completing the square.

a) f (x) = x² + 6x + 5

b) f (x) = 3x² – 12x – 9

Ex: Consider the function y = 3x² + 12x + 8.

a) Complete the square to determine the vertex and the maximum or minimum value of the function.

b) Check your result using the formula p = -b/2a c) Graph the quadratic function in the grid provided.

Ex: The student council at a high school is planning a fundraising event with a professional

photographer taking portraits of individuals or groups. The student council gets to charge and keep a session fee for each individual or group photo session. Last year, they charged a $10 session fee and 400 sessions were booked. In considering what price they should charge this year, student council members estimate that for every $1 increase in the price, they expect to have 20 fewer sessions booked.

a) Write a function to model this situation.

b) What is the maximum revenue they can expect based on these estimates. What session fee will give that maximum?

c) How can you verify the solution?

d) What assumptions did you make in creating and using this model function?