9-1: Quadratic Graphs and Their Properties

Objective: To explore graphs of quadratic functions

Warm – Up:

Evaluate each expression for 4x , 3y , and 2z .

1.

2xy 2.

2xz

3.

21

2x y 4.

21

3y z

Graph each equation. For 6 and 7, make a table of values.

5. 2 3y x 6. 1y x 7.

2 2y x

Exploration: Comparing Quadratic Graphs

1) Graph the equations

2y x and

23y x on the same coordinate plane.

2) Describe how the graphs are alike and different.

3) Predict how the graph of

21

3y x will be similar to and different from

2y x .

4) Graph

21

3y x . Were your predictions correct?

5) Graph the equations

22y x and

22y x on the same coordinate plane.

6) Describe how the graphs are alike and different.

7) How can you tell if the graph opens up or down?

8) Graph the equations

2 3y x and

2 3y x on the same coordinate plane.

9) How does adding or subtracting a number affect the graph?

x

y

x

y

x

y

For the quadratic function,

2y ax bx c :

If a is positive, the parabola opens up.

If a is negative, the parabola opens down.

If a is larger than 1, the parabola is narrower than

2y x .

If a is smaller than 1, the parabola is wider than

2y x .

quadratic parent function: The simplest quadratic function,

2y x

vertex: The lowest point of a parabola that opens up (minimum) or the highest point of a parabola

that opens down (maximum)

axis of symmetry: The line passing through the vertex that divides the parabola into two mirror

image halves

Example 1:

Identify the vertex of each graph. Tell whether it is a minimum or maximum. State the axis of

symmetry of the graph.

a) b)

Example 2:

Order the quadratic functions

2( ) 4f x x ,

21( )

4f x x , and

2( )f x x from the widest to the

narrowest graph.

Example 3:

Make a prediction about the graph of each quadratic.

(Refer to the opening direction, the width as compared to

2y x , and the coordinates of the vertex.)

a)

22y x b)

218

5y x

Example 4:

A child drops a pebble from a height of 75 feet above a lake. The function

2( ) 16 75h t t gives

the height h of the pebble, in feet, after t seconds.

a) Graph the function using a table of values.

b) At about what time does the pebble hit the water?

c) How far has the pebble fallen from time t = 0 to t = 1?

d) Does the pebble fall the same distance from time t = 1 to t = 2

as it does from t = 0 to t = 1? Explain your reasoning.

Closure Question:

In a quadratic function

2y ax c , how do a and c affect the graph?

y

9-2: Standard Form of a Quadratic Function

Objective: To graph quadratic functions of the form

2y ax bx c

Warm – Up:

Three parabolas are shown with their equations below.

1. Identify the vertex for each parabola.

2. Draw the axis of symmetry on each graph. Write the equation for the axis.

3. Calculate the ratio

2

b

a

for each equation.

22 4y x

22 4 2y x x

22 8 4y x x

How is the ratio calculated in part 3 related to the axis of symmetry for each graph?

axis of symmetry: vertical line

2

bx

a

vertex: ordered pair ,

2 2

b bf

a a

Increasing Function: As the x-value increases, the y-value increases (positive rate of change)

Decreasing Function: As the x-value increases, the y-value decreases (negative rate of change)

Example 1:

Consider the function

2( ) 5 20 1f x x x .

a) State whether the graph opens up or down.

b) Write the equation of the axis of symmetry.

c) Calculate the coordinates of the vertex.

Graphing a Quadratic Function:

1) Calculate the equation of the axis of symmetry.

2) Calculate the coordinates of the vertex.

3) Make a table of values, using two x-coordinates to the left and right of the vertex.

4) Plot the 5 points and connect them to form a parabola.

For examples 2- 4, calculate the equation of the axis of symmetry and the coordinates of the vertex

to graph each quadratic function.

Example 2:

2( ) 2 3f x x

Example 3:

2( ) 4 1f x x x

Example 4:

2( ) 3 6 9f x x x

Example 5:

A ball is thrown into the air with an initial upward velocity of 48 ft/sec.

Its height h, in feet, after t seconds is given by the function

2( ) 16 48 4h t t t .

a) In how many seconds will the ball reach its maximum height?

b) What is the ball’s maximum height?

Closure Question:

Describe some characteristics of the graph

2( ) 2 8 1f x x x .

x

y

x

y

x

y

Vertex Form of a Quadratic Function

Objective: To write quadratic equations in vertex form using the method of completing the square

To identify key features of the graph of a quadratic function from vertex form

Warm – Up:

Solve each quadratic equation by completing the square.

1.

2 4 21x x 2.

2 18 17 0x x

Write an equation in point-slope form for the line through the given point that has the given slope.

3. (9, 5) ; m = 6 4. ( 7, 2) ; m = 3

* Vertex Form: The equation 2( ) y a x h k represents a parabola with vertex ( , )h k .

Recall that “a ” determines the opening direction and width of the parabola.

Example 1:

For each quadratic equation, identify the opening direction, the width compared to

2y x , the

coordinates of the vertex, and whether the vertex is a maximum or minimum.

a)

25( 2) 1y x b) 29( 4)y x c)

21( 3) 7

6y x

Example 2:

Write a quadratic equation for a parabola with the given vertex.

a) (8, 6) b) (0, 5) c) ( 1, 9)

Are these the only quadratic equations with graphs with these vertices? Explain your reasoning.

For examples 3-5, use the method of completing the square to write each quadratic equation in

vertex form and identify the vertex. Then determine whether the vertex is a maximum or minimum

and state its value.

Example 3:

2 6 7y x x

Example 4:

2 2 8y x x

Example 5:

2 10 31y x x

Closure Question:

Explain how the graph of

27( 1) 4y x compares to

2y x .

(Refer to the opening direction, the width, the coordinates of the vertex, and whether the vertex is a

maximum or minimum.)

Factored Form of a Quadratic Function

Objective: To write quadratic equations in factored form using the method of factoring

To identify key features of the graph of a quadratic function from factored form

Warm – Up:

Solve each quadratic equation by factoring.

1.

2 2 35 0x x 2.

23 14 24 0x x

Given the x-intercepts of the graph of a quadratic function, write the possible factors.

3. 1x , 9x 4. 3x , 8x

* Factored Form: The equation 1 2( )( )y a x x x x represents a parabola where 1x and 2x are

x-intercepts or zeros of the graph.

Recall that “a ” determines the opening direction and width of the parabola.

Example 1: For each quadratic function, identify the zeros, determine the axis of symmetry, calculate the vertex,

and graph the function using the zeros and vertex.

a) ( ) ( 4)( 2)f x x x b) ( ) 5 ( 6)f x x x

Example 2:

For each graph, identify the zeros and write a quadratic function in factored form. Use the given

“a” value to complete the equation.

a) b)

1a

1

4a

Example 3:

Factor each quadratic function, identify the zeros, determine the axis of symmetry, calculate the

vertex, and graph the function using the zeros and vertex.

a) 2( ) 2 15f x x x b) 2( ) 4 20 11f x x x

Closure Question:

Where is the vertex located in relation to the x-intercepts of the graph of a quadratic function?

Explain your reasoning.

Key Features of Graphs

Objective: To identify key features of graphs

Warm – Up:

1. State whether the graph of

2( ) 5 7f x x has a maximum or minimum. Explain your reasoning.

2. Order the quadratic functions from the widest to the narrowest graph:

2( ) 8f x x

21( )

6f x x

2( )f x x

2( ) 3f x x

3. Without graphing, predict how the graph of

21( ) 9

4f x x compares to

2( )f x x .

Domain: The set of all possible input values

Range: The set of all possible output values

Maximum: The largest y-value on the graph of a function

Minimum: The smallest y-value on the graph of a function

Axis of Symmetry: The line that divides a figure into two mirror image halves

End Behavior: The behavior of a graph of f(x) as x approaches positive and negative infinity

Increasing Function: As the x-value increases, the y-value increases (positive rate of change)

Decreasing Function: As the x-value increases, the y-value decreases (negative rate of change)

Example 1:

For the graph of ( ) 4f x x , identify the following key features.

a) domain:

b) range:

c) x-intercept(s):

d) y-intercept:

e) maximum:

f) minimum:

g) axis of symmetry:

h) end behavior:

i) Label the graph where the function is increasing and decreasing.

Example 2:

For the graph of

31 9( ) 9

6 2f x x x , identify the following key features.

a) domain:

b) range:

c) x-intercept(s):

d) y-intercept:

e) maximum:

f) minimum:

g) axis of symmetry:

h) end behavior:

i) Label the graph where the function is increasing and decreasing.

Example 3:

The point (4, 5) is the vertex of the graph of a quadratic function. The points (1, 0) and ( 8, 6 ) are

also on the graph of the function. Complete the graph of this quadratic function by first finding two

additional points on the graph. Then identify the following key features.

a) domain:

b) range:

c) x-intercept(s):

d) y-intercept:

e) maximum:

f) minimum:

g) axis of symmetry:

h) end behavior:

i) Label the graph where the function is increasing and decreasing.

Closure Question:

Given the function

2( ) 7 1f x x , for which interval is the rate of change positive and which

interval is the rate of change negative?

Translations of Functions

Objective: To translate the graphs of quadratic, absolute value, exponential, and square root functions

Warm – Up: Graph each parent function by making a table of values.

1. 2y x 2. y x

3. 2xy 4. y x

Identify the vertex for each quadratic equation and describe the direction that the parabola is shifted.

5.

2( 9) 4y x 6.

2( 7) 5y x

Transformation: A change in the position or size of a graph

Type 1 - A translation is a transformation that shifts a graph horizontally, vertically, or both.

The result is a graph of the same size and shape, but in a different position.

Example 1:

Describe each transformation.

a) 3y x b) 2 6xy c)

2( 8) 1y x

Example 2:

Write an equation for each transformation using the given parent functions.

a) 2xy b) y x c) y x

translation 7 units up translation 4 units right translation 9 units left

and 2 units down

Transformation Form of Parent Functions

2( )y a x h k y a x h k 2x hy a k y a x h k

where h is the horizontal translation and k is the vertical translation

x

y

x

y

x

y

x

y

Example 3:

Using the parent function as a guide, describe the transformation and graph each function.

a) 2 5xy

b)

2( 2) 3y x

c) 1 7y x

Example 4: Describe the transformation and write an equation for each function. a) b) c)

Closure Question:

Describe the domain and range of the graph

2( 1) 4y x .

x

y

x

y

x

y

Scale Changes and Reflections of Functions

Objective: To vertically stretch, shrink, and reflect the graphs of quadratic, absolute value,

exponential, and square root functions

Warm – Up: Use the quadratic functions 2y x , 23y x , and 21

3y x to answer the questions below.

1. Order the functions from the widest to the narrowest.

2. Does the coefficient “a” affect the x or y values of the graph? Explain your reasoning.

3. What values of “a” make a parabola wider and what values make it narrower than 2y x ?

Why do you think this happens?

Transformation: A change in the position or size of a graph

Type 2 - vertical scale change and reflection over the x-axis

o A vertical scale change is a transformation that stretches or shrinks a graph in the

vertical direction.

o A reflection over the x-axis is a transformation that produces the mirror image of

a graph with the x-axis as the line of symmetry.

Example 1:

Describe each transformation.

a) 8y x b) 1

25

xy c) 6y x

Example 2:

Write an equation for each transformation using the given parent functions.

a) 2y x b) y x c) 2xy

vertical shrink of 1

9 vertical stretch of 7 vertical shrink of

1

4

reflection over the x-axis reflection over the x-axis

Transformation Form of Parent Functions

2( )y a x h k y a x h k 2x hy a k y a x h k

where a is the vertical scale change and a negative a value indicates a reflection over the x-axis

Example 3:

Using the parent function as a guide, describe the transformation and graph each function.

a) 3y x

b) 1

2y x

c) 6 2xy

Closure Question: Without making a table of values, sketch the following graphs and label each function.

2( )f x x

2( ) 4g x x

2( ) 0.5h x x

2( ) 9k x x

x

y

x

y

x

y

Combining Transformations of Functions

Objective: To translate, stretch, shrink, and reflect the graphs of quadratic, absolute value, exponential,

and square root functions

Warm – Up: Write an equation for each transformation using the given parent functions.

1. y x 2. 2xy

translation 2 units right vertical shrink of 1

8

translation 9 units down reflection over the x-axis

Using the parent function as a guide, describe the transformation and graph each function.

3.

2( 3) 5y x 4. 6y x

Example 1:

Describe each transformation.

a)

12 8

9

xy b) 2 5y x c)

24( 7) 3y x

Example 2:

Write an equation for each transformation using the given parent functions.

a) y x b) 2xy c) y x

translation 9 units left translation 6 units down translation 1 unit right

vertical stretch of 7 vertical shrink of 1

3 translation 5 units up

reflection over the x-axis vertical stretch of 8

reflection over the x-axis

x

y

x

y

Transformation Form of Parent Functions

2( )y a x h k y a x h k 2x hy a k y a x h k

where h is the horizontal translation, k is the vertical translation,

a is the vertical scale change, and a negative a value indicates a reflection over the x-axis

Example 3:

Using the parent function as a guide, describe the transformation and graph each function.

a) 23( 4)y x

b) 1

5 62

y x

c) 4 1 7y x

Closure Question:

Compare the graphs of the functions

2( ) 3( 8) 6f x x and

2( ) 9( 5) 1g x x . Describe

two ways these graphs are similar and two ways they are different.

x

y

x

y

x

y