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Graphing Quadratic Function

Standard Form:

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Graphing Quadratic Function

The axis of symmetry

divides the parabola into

two congruent halves.

Each half is a reflection

of the other half.

Axis of symmetry

intersects the vertex.

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Graphing Quadratic Function

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Graphing Quadratic Function

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Graphing Quadratic Function

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Graphing Quadratic Function

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Graphing Quadratic Function

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Graphing Quadratic Function

How would we get the y-coordinate of the vertex?

� � � �� � 4x � 3

� �2 � ��2���4��2� � 3

� �2 � 4 � 8 � 3

� �2 � �7

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Graphing Quadratic Function Use the vertex and two other x-values to graph the function.

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Graphing Quadratic Function

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Graphing Quadratic Function

Vertex Form:

Standard Form:

Intercept Form:

� � ��� � �����

� � ��� � �� � �

� � ��� � ���� � ��

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Graphing Quadratic Function

Graph � � ��� � 2��� � 4�

� � ��� � ���� � ��

� � �1

� � �2

� � 4 Vertex: � � ��1 � 2��1 � 4�

� � ��3���3�

� � 9 �1,9�

Another Point: � � ��2 � 2��2 � 4�

� � ��4���2�

� � 8 �2,8�

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Graphing Quadratic Function

� � �� � 1��� � 5� � � �2�� � 1��� � 5�

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Max and Min Value

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Max and Min Value

Determine whether the function has a maximum or

minimum value.

State the maximum or minimum value.

The maximum value of the function is the y-coordinate of the vertex.

� � �1 Negative ‘a’ means there will be max. value

� � ����

� � ������ �

� � ����!

� � 1.5

� � � �4�� � 12� � 18

� 1.5 � �4 1.5 � � 12�1.5� � 18

� 1.5 � �9 � 18 � 18

� 1.5 � 27

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Max and Min Value

Does the function have a maximum or minimum value?

State the maximum or minimum value of the function.

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Max and Min Value Researchers conducted an experiment to determine

temperatures at which people feel comfortable. The

percent y of test subjects who felt comfortable at temperature x (in degrees Fahrenheit) can be modeled by:

� � �3.678�� � 527.3� � 18,807

What temperature made the greatest percent of test

subjects comfortable? At that temperature, what percent

felt comfortable?

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Max and Min Value

� � �4.290�� � 612.6� � 21,773

The equation in the previous problem is based on

preferences of both male and female test subjects.

Researchers also analyzed data for males and females

separately and obtained the equations below:

Males:

Females: � � �6.224�� � 908.9� � 33,092

What was the most comfortable temperature for the males?

For the females?

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Max and Min Value The Golden Gate Bridge in San Francisco has two towers that rise

500 feet above the road and are connected by suspension cables as

shown. Each cable forms a parabola with equation:

� � �!%&'

�� � 2100���8

where x and y are

measured in feet.

a. What is the distance

d between the two towers?

b. What is the height l above the road of a

cable at its lowest

point?

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Max and Min Value

Although a football field appears to be flat, its surface is actually

shaped like a parabola so that rain runs off to either side. The cross

section of a field with synthetic turf can be modeled by:

� � �0.000234 � � 80 � � 1.5

where x and y are measured in

feet. What is the field’s width?

What is the maximum height of

the field’s surface?

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Factoring Greatest Common Factor (GCF)

2�� � �

2 ∙ 1 ∙ � ∙ � � 1 ∙ 1 ∙ �

1��2� � 1�

)*+ � 1�

��2� � 1�

2� � ��

2 ∙ 1 ∙ � ∙ � ∙ � ∙ � � 1 ∙ 1 ∙ � ∙ � )*+ � 1��

1���2�� � 1�

1���2�� � 1�

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Factoring Greatest Common Factor (GCF)

4�� � 2�

2 ∙ 2 ∙ � ∙ � � 2 ∙ 1 ∙ �

2��2� � 1�

)*+ � 2�

6� � 3��

3 ∙ 2 ∙ � ∙ � ∙ � ∙ � � 3 ∙ 1 ∙ � ∙ � )*+ � 3��

3���2�� � 1�

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Factoring Greatest Common Factor (GCF)

8�� � 4� � 2

2 ∙ 2 ∙ 2 ∙ � ∙ � � 2 ∙ 2 ∙ � � 2 ∙ 1 )*+ � 2

2�4�� � 2� � 1�

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Factoring

Factor:

8�� � 4� 12�� � 9� 17�� � 16�

6�� � 4� � 2 24�� � 12� � 3

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Factoring

�� � �� � � � �� �,��� � -�

Factoring a Trinomial of the Form: �� � �� � �

� �� ��, � -�� �,-

Steps:

1. Find the factors of c.

2. Determine which factors of c add to b.

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Factoring Factoring a Trinomial of the Form:

�� � 12� � 28

Factors of -28 (m,n) -1, 28 1, -28 -2, 14 2, -14 -4, 7 4, -7

Sum of factors (m+n) 27 -27 12 -12 3 -3

Therefore , � 2 - � �14

�� � 12� � 28 � �� � 2��� � 14�

�� � �� � �

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Factoring

�� � � � 2 �� � 6� � 5

�� � 6� � 8 �� � 4� � 12

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Factoring

��� � �� � � � ��� � ,��.� � -�

Factoring a Trinomial of the Form: ��� � �� � �

� �.�� ���- � .,�� � ,-

Steps:

1. Multiply a times c.

2. Find factors of ac.

3. Determine which factors of ac add to b.

4. Rewrite b term by summing the factors of the ac term.

5. Group first two and last two terms.

6. Factor out GCF from two groups.

7. Write the product of two binomials.

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Factoring Factoring a Trinomial of the Form: ��� � �� � �

3�� � 17� � 10

Factors of 30 (m,n) -1, -30 1, 30 -2, -15 2, 15 -3, -10 3, 10 -5, -6 5, 6

Sum of factors

(m+n)

-31 31 -17 17 -13 13 -11 11

� ∙ � � 3 ∙ 10 � 30

3�� � 15� � 2� � 10

3�� � 15� � ��2� � 10�

3� � � 5 � 2�� � 5�

�3� � 2� � � 5

3�� � 2� � 15� � 10

�3���2�� � ��15� � 10�

� 3� � 2 � 5�3� � 2�

�� � 5��3� � 2�

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Factoring

2�� � � � 3 3�� � 8� � 4

2�� � 9� � 4 5�� � 7� � 2

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Factoring – Special Cases

�� � �� � �� � ���� � ��

Factoring a Difference of Two Squares:

Where does the third term go?

�� � ���� � ��

�� � �� � �� � ��

�� � �� Difference of Two Squares!

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Factoring – Special Cases

�� � 4

Example:

Factor

�� � 2� �� � �� � �� � ���� � ��

� � � � � 2

�� � 2��� � 2�

A different approach:

�� � 4 �� � �� � 4 � � 0

Need factors of -4 that add to zero: �2, 2

�� � 2��� � 2�

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Factoring – Special Cases

�� � 1 �� � 9 �� � 49

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Factoring – Special Cases

4�� � 25

Example:

Factor

�2����5� �� � �� � �� � ���� � ��

� � 2� � � 5

�2� � 5��2� � 5�

A different approach:

4�� � 25 4�� � �� � 25 � � 0

Need factors of -100 that add to zero: �10, 10

4�� � 10� � 10� � 25

�4���10�� � �10� � 25�

2��2� � 5� � 5�2� � 5�

�2� � 5��2� � 5�

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Factoring – Special Cases

9�� � 16 4�� � 36 16�� � 9

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Factoring – Special Cases Factoring GCF First:

5�� � 20

5��� � 4�

5�� � 2��� � 2�

6�� � 15� � 9

3�2�� � 5� � 3�

3 �2�� � 3� � 2� � 3�

3 �2�� � 3�� � �2� � 3�

3 ��2� � 3� � 1�2� � 3�

3 �2� � 3��� � 1�

3�2� � 3��� � 1�

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Factoring – Special Cases

5�� � 5� � 10 18�� � 2

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Solving Quadratic Equations

�� � 3� � 18 � 0

Use factoring to help solve quadratic equations:

� � 6 � � 3 � 0 Factor

Zero Product Property:

/�01 � 0, 2�3-342�350 � 0651 � 0

� � 6 � 0 � � 3 � 0 or Zero Product Property

� � �6 or � � 3 Solve for x

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Solving Quadratic Equations

�� � � � 6 � 0 3�� � 10� � 3 � 0

5�� � 30� �� � 14� � �49

1. 2.

3. 4.

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Using a Quadratic Model You have made a rectangular stained glass

window that is 2 feet by 4 feet. You have 7

square feet of clear glass to create a

border of uniform width around the

window. What should the width of the

border be?

789:;<=;8>98 � ?;@:A789: � 789:;<BCD>;E

F � �G � HI��H � HI� � �H ∗ G�

F � �G � HI��H � HI� � K

F � K � KI � GI � GIH � K

F � LHI � GIH

M � GIH � LHI � F

M � GIH � HI � LGI � F

M � �GIH � HI� � �LGI � F�

M � HI�HI � L� � F�HI � L�

M � �HI � F��HI � L�

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Using a Quadratic Model You have made a rectangular stained glass

window that is 2 feet by 4 feet. You have 7

square feet of clear glass to create a

border of uniform width around the

window. What should the width of the

border be?

M � GIH � HI � LGI � F

M � �GIH � HI� � �LGI � F�

M � HI�HI � L� � F�HI � L�

M � �HI � F��HI � L� HI � L � M or HI � F � M

HI � L

I � M.5

HI � �F

I � �N. O

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Using a Quadratic Model

You have just planted a rectangular flower

bed of red roses in a park near your home.

You want to plant a border of yellow roses

around the flower bed as shown. Since

you bought the same number of red and

yellow roses, the areas of the border and

inner flower bed will be equal. What

should the width x of the border be?

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Finding Zeros of a Quadratic Zeros of a function is another name for the x-intercepts or the

solutions of the function.

� � ��� � 2��� � 4�

��2, 0� �4, 0�

X – Intercepts:

Zeros:

�2�-P4

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Finding Zeros of a Quadratic Find the Zeros:

� � �� � � � 6

� � �� � 2��� � 3�

Zeros:

�2�-P3

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Finding Zeros of a Quadratic

� � �� � 3� � 2 � � �� � 7� � 12

� � 2�� � 2� � 24 � � 3�� � 8� � 4

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Solve Quadratic by Square Root

Radical Sign

Q

Square Root

Expression is called a radical

“s” is called a radicand

�� � 4

A positive number has two square roots.

2�-P � 2 �2��� 4�-P��2��� 4

� � R 4

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Solve Quadratic by Square Root

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Solve Quadratic by Square Root

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Solve Quadratic by Square Root

“rationalize the denominator”

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Solve Quadratic by Square Root

Simplify the expression:

49

16

25

12 45 3 ∙ 27

7

9

5

2

1

3

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Solve Quadratic by Square Root

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Solve Quadratic by Square Root

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Solve Quadratic by Square Root

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Solve Quadratic by Square Root

The height of an object in free-fall can be modeled by the

following equation:

� � �162� � �0

�

2

�0

Object’s final height (ft)

Elapsed time (s)

Object’s initial height (ft)

Assumptions:

1. Air resistance is

negligible.

2. Near the Earth’s surface.

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Solve Quadratic by Square Root

A stunt man working on the set of a movie is to fall out of a

window 100 feet above the ground. For the stunt man’s safety, an

air cushion 26 feet wide by 30 feet long by 9 feet high is positioned

on the ground below the window.

a. For how many seconds will the stunt man fall before he reaches

the cushion?

b. A movie camera operating at a speed of 24 frames per second

(fps) records the stunt man’s fall. How many frames of film show

the stunt man falling?

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Solve Quadratic by Square Root

� � �162� � �0

9 � �162� � 100

+4-�.S34T�2?

/-424�.S34T�2?

9

100

�91 � �162�

91

16� 2�

91

16� 2

2 V 2.4Q3�6-PQ

�6W,�-��5�,3Q6��4.,?

24�5�,3Q

1Q3�6-P∙2.4Q3�6-PQ

1

��6X257�5�,3Q

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Solve Quadratic by Square Root

At an engineering school, students are challenged to design a

container that prevents an egg from breaking when dropped from

a height of 50 feet. Write an equation giving a container’s height h (in feet) above the ground after t seconds. How long does the container take to hit the ground?

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Solve Quadratic by Square Root Felix Baumgartner recently made the highest skydive ever,

jumping from a distance of 128,100 feet above the earth’s surface.

It was estimated that he went approximately 833.9 mph. If he did

not open his shoot and air resistance was neglected, how long

would it take Felix to reach the Earth’s surface jumping from this

height?

If his velocity could be calculated by the equation:

Y � �322

where v is the final velocity in feet per second (ft/s) and t is the elapsed time in seconds (s)

What would have been his final velocity knowing how long he fell?

Give answer in mph. Is this answer different then his actual

speed? Why or why not?

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Solve Quadratic by Square Root Will a penny dropped from the top of the Empire State Building

kill someone if they get hit in the head? Could the penny

embedded itself into the concrete?

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Solve Quadratic by Square Root Project:

Write a one page paper describing:

1. What terminal velocity is.

2. How terminal velocity can be calculated.

3. Calculate the terminal velocity of any object of your choice.

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Solve Quadratic by Square Root

The equation � � 0.019Q�

gives the height h (in feet) of the largest ocean waves when the

wind speed is s knots. How fast is the wind blowing if the

largest waves are 15 feet high?

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Solve Quadratic by Square Root

� � �0�2ZP� 3

.W2

�

For a bathtub with a rectangular base, Torricelli’s law implies

that the height h of water in the tub t seconds after it begins draining is given by:

where l and w are the tub’s length and width, d is the diameter of the drain, and h0 is the water’s initial height. (All measurements are in inches.) Suppose you completely fill a tub with water. The tub is 60

inches long by 30 inches wide by 25 inches high and has a drain with

a 2 inch diameter.

a. Find the time it takes the tub to go from being full to half-full.

b. Find the time it takes the tub to go from being half-full to empty.

c. What conclusions can you make about the speed the water drains?

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula

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Quadratic Formula