Forms & Characteristics Quadratic Functions

​Quadratic Functions

​Key Terms:

Examples!

Key Features of this Quadratic

• Domain: All Real Numbers or −∞ < x < ∞

• Range: [-4, ∞) or −4 ≤ y < ∞

• X-intercept(s)/Zeros/Roots: (-3, 0) & (1, 0)

• Y-intercept:(0, -3)

• Vertex/Minimum: (-1,-4)

• Axis of Symmetry: x = -1

• Interval of Increase: x > -1

• Interval of Decrease: x < -1

• End Behavior: 

  • as x → −∞ , f(x) → ∞

  • as x → ∞, f(x) →  ∞

​.

​.

​.

​.

D​omain Definition

​

the domain is the set of inputs accepted by the function

​

our x-values make up the domain

​

We can scan the x-axis from left to right to find our domain.

Range Definition

​The range is our output values accepted by the function

​

therefore, y-values make up the range ​

Scan the y-axis from the bottom to the top to determine range.

Domain, Range, and End Behavior

Please write this review information down in your notebook. Draw the pictures. They do NOT have to be perfect, just a sketch. Follow the steps from the video.

Drag and Drop

What is the Domain?



​

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<​ ​

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<​

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Drag these tiles and drop them in the correct blank above

Drag and Drop

What is the Range?



​

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<​ ​

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​≤

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Drag these tiles and drop them in the correct blank above

Drag and Drop

What is the End Behavior? 

As x → −∞ , f(x) → ​ ​

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As x → ∞, f(x) →  ​ ​

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​

Drag these tiles and drop them in the correct blank above

Hotspot

Select the y-intercept.

​

Fill in the Blanks

Multiple Choice

How many x-intercepts? 

​

• 2

• 1

• 0

Hotspot

Select the x-intercepts.

​

Fill in the Blanks

Dropdown

What are the zeros? 

Hint: Enter the values from Left to Right.



​ x = ​

Select answer

and x =​

Select answer

Hotspot

Select the vertex.

​

Fill in the Blanks

Multiple Choice

What is the vertex? 

• Maximum

• Minimum

Drag and Drop

Use the graph above and find the axis of symmetry. 



x =​

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Drag these tiles and drop them in the correct blank above

-1

-3

-2

-4

-5

0

1

2

3

Graphing

Use the graph above and drow the axis of symmetry. 

Hint: Let one point be the vertex and the other point fall on the x-axis.

Drag and Drop

Use the graph above and find the intervals. 



Interval of Increase: ​

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Interval of Decrease:​

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Drag these tiles and drop them in the correct blank above

x < -1

x > -1

Poll

How do you feel about characteristics of quadratic functions?

• I'm clueless

• I need more practice

• I got this!

Key Features of this Quadratic (Real World)

• Domain: [0,40]

• Range: [0,800]

• Maximum: (20,800)

• Zeros: (0,0) and (40,0)

• Interval of Increase: 0 < x < 20

• Interval of Decrease 20 < x < 40

Fill in the Blanks

Dropdown

The vertex for D(x) is (4, 2). The dolphin reaches a maximum height of 2 meters at 4 seconds.

What does the vertex of B(x) mean in context?



The Bird reaches a ​

Select answer

height of ​

Select answer

meters at ​

Select answer

seconds

Drag and Drop

The axis of symmetry of D(x) is x = 4. This is the x value that splits the graph into 2 equal pieces.


What is the axis of symmetry of B(x)?



x =​

Placeholder

Drag these tiles and drop them in the correct blank above

-1

0

1

2

3

4

5

6

7

Hotspot

The x-intercepts (or zeros or roots) of a function are the point(s) where the function crosses or touches the x-axis. The x-intercepts of B(x) are (1, 0) and (5, 0). This means that the height of the bird above sea level was 0 meters at 1 second and 5 seconds.

What are the x-intercepts of D(x)?

​

Fill in the Blanks

Dropdown

The x-intercepts of B(x) are (1, 0) and (5, 0). This means that the height of the bird at sea level was 0 meters at 1 second and 5 seconds.


What does the x-intercepts of D(x) mean in context?



This means that the height of the dolphin at sea level was 0 meters at ​

Select answer

second and ​

Select answer

seconds.

Hotspot

The y-intercept of a function is the point that crosses the y-axis. This is also called the initial (start) value. The y-intercept of D(x) is (0, -6). This means at the start the dolphin was 6 meters below sea level.

What is the y-intercepts of B(x)?

​

Fill in the Blanks

Dropdown

The y-intercept of D(x) is (0, -6). This means at the start the dolphin was 6 meters below sea level.


What does the y-intercept of D(x) mean in context?



This means at the start the bird was ​

Select answer

meters ​

Select answer

sea level.

Dropdown

Function D(x) has an interval of increase when 0 ≤ x < 4 because the dolphin’s height is going up between 0 and 4 seconds.

What are the intervals of increase for B(x)?

Function B(x) has an interval of increase when

​

Select answer

≤ x < ​

Select answer

because the height of the bird is going up between ​

Select answer

seconds and ​

Select answer

seconds.

Dropdown

Function D(x) has an interval of decrease of

when 4 < x ≤ 7 because the height of the dolphin is going down between 4 seconds and 7 seconds.

What are the intervals of decrease for B(x)?

Function B(x) has an interval of decrease of

when ​

Select answer

< x ≤ ​

Select answer

because the height of the bird is going down between ​

Select answer

seconds and ​

Select answer

seconds.

Drag and Drop

The domain of D(x) is 0 ≤ x ≤ 7 (0 to 7 seconds)
What are the domain of B(x)?

​

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​ ≤​ ​

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≤​ ​

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(​

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to ​

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seconds)

Drag these tiles and drop them in the correct blank above

0

6

7

1

3

4

2

5

Drag and Drop

The range of D(x) is –6 ≤ y ≤ 2 (6 meters below sea level to 2 meters above sea level).
What are the range of B(x)?

​ ​

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≤ ​ ​

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≤ ​ ​

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​

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meters below sea level to ​

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meters above sea level

Drag these tiles and drop them in the correct blank above

-4

5

-2

-1

1

2

4

0

Categorize

Options (12)

The y-intercept of b(x) is (0, 6)

b(4) = 10

The height of the banana is decreasing on the interval 2.5 < x ≤ 6.

The vertex of the function is at (3, 12).

The banana reaches a maximum height of approximately 12.25 feet.

The banana is launched from a cannon 6 feet above the ground.

The interval of increase of the function is 0<x<4.

When the banana is 5 meters from the cannon, it is 8 meters above the

ground.

The banana has reached the same height as the cannon when it was 5 feet away from the cannon.

b(x) has a domain of 0 ≤ x ≤ 6.

The banana reaches the ground 5 feet from the cliff.

The function has an axis of symmetry at x = 2.5.

Sort the following statements into two categories(True or False).

True

False

Labelling

Closing:

Label the following characteristics on the graph.

Drag labels to their correct position on the image

vertex

maximum

x-intercept

minimum

y-intercept