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Section 2.2Quadratic Functions

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Graphs of Quadratic Functions

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Graphs of Quadratic FunctionsParabolas

x

y

x

y

MinimumVertex

Axis of symmetry Maximum

2( )f x ax bx c

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2

Quadratic functions are any function of the form

f(x)=ax +bx+c where a 0, and a,b and c are

real numbers. The graph of any quadratic

function is called a parabola. Parabolas are

shaped like cups. Para

bolas are symmetic with

respect to a line called the axis of symmetry.

If a parabola is folded along its axis of symmetry,the two halves match exactly.

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Graphing Quadratic Functionsin Standard Form

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Seeing the Transformations

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Using Standard Form

2

finding the y intercept let x=0

y=-2(0-3) 8 y=-10 (0,-10)

x

y2( ) 2( 3) 8f x x

( , ) V(3,8)vertex h k

axis of symmetry x=3

2

2

2

2

finding the x intercept, let y=0

0=-2(x-3) 8

8 2( 3)2 2

4 ( 3)

4 ( 3)

2 3

3 2 , (5,0) (1,0)

x

x

x

x

x

a

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Example

Graph the quadratic function f(x) = - (x+2)2

+ 4.

x

y

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Example

Graph the quadratic function f(x) = (x-3)2

- 4

x

y

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Graphing Quadratic Functionsin the Form f(x)=ax2=bx+c

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2

We can identify the vertex of a parabola whose equation is in

the form f(x)=ax +bx+c. First we complete the square.

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Using the form f(x)=ax2+bx+c

2

Finding y intercept

y=0 2 0 1

1 (0,1) y intercepty

y

2( ) 2 1 a=1, b=2, c=1f x x x

2

-b -2, x= 12a 2 2 1

( 1) ( 1) 2( 1) 1 0 V(-1,0)

bVertex f a

f

Axis of symmetry x=-1

2

Finding x intercept

0=x 2 1

0 ( 1)( 1)

1 0

1 (-1,0) x intercept

x

x x

x

x

a>0 so parabola has aminimum, opens up

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Example

Find the vertex of the function f(x)=-x2

-3x+7

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Example

Graph the function f(x)= - x2

- 3x + 7. Use thegraph to identify the domain and range.

x

y

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Minimum and Maximum Valuesof Quadratic Functions

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Example

For the function f(x)= - 3x2

+ 2x - 5Without graphing determine whether it has aminimum or maximum and find it.

Identify the functions domain and range.

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Graphing Calculator Finding theMinimum or Maximum

Input the equation into Y=

Go to 2nd Trace to get Calculate. Choose #4 for

Maximum or #3 for Minimum.

Move your cursor to the left (left bound) of therelative minimum or maximum point that you

want to know the vertex for. Press Enter.

Then move your cursor to the other side of thevertex the right side of the vertex when it asks

for the right bound. Press Enter.When it asks to guess, you can or simply pressEnter.

The next screen will show you the coordinates ofthe maximum or minimum.

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Applications of QuadraticFunctions

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Quadratic Regression on the Graphing Calculator

M Kwh

1 8.79

2 8.16

3 6.25

4 5.395 6.78

6 8.61

7 8.788 10.96

A consumer decided to record their electric use

in hundreds of kilowatt hours. The first columnfor the month of the year. Put the data into List1 &

List2 in the graphing calculator.

To do that Press STAT,

then 1 for edit. Type in

numbers.

2

Press STAT, move the

cursor to the right to CALC,

then press 5 for QuadReg.

The quadratic equation that

describes power use is

y=.29x 2.35 10.97x

More on thenext slide.

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Quadratic Regression on the Graphing Calculator

To see the scatter plot of these data pointspress 2nd Y= to get STAT PLOT. Press

ENTER on #1. If Plot 2-3 are ON, thenchange those to OFF. You can turn all plotsoff by pressing #4. Then return to thisscreen and press #1 to turn this plot on.

By pressing GRAPH you will getthe graph that you see at left.

This is the Plot1 Screen. Press ENTER

on the word On. Cursor down and choosethe style of graph that you want. The firstis a scatterplot. The XList should be L1,and YList L2. Choose one of the marks foryour graph. For L1 or L2 press 2nd 1 or2nd 2.

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Example

You have 64 yards of fencing to enclose arectangular region. Find the dimensions of the

rectangle that maximize the enclosed area. What isthe maximum area?

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Graphing our previous problem.

Change the viewing rectangle to [0,32,4] and [0,260,20]

Why are we using this Window?

What is the maximum point on the graph? Use your

graphing calculator's Maximum function under Calculate

to find the maximum.

Graphing Calculator

Problem continued on the next slide

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Graphing Calculator- continued

ndDoes the Table (2 Graph) show the same maximum?

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(a)

(b)

(c)

(d)

2

Find the coordinates of the vertex for

the parabola defined by the given equation.f(x)=2(x-4) 5

(2,5)

( 2, 5)

( 4,5)

(4,5)

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(a)(b)

(c)

(d)

2

Use the graph of the parabola to

determine the domain and range ofthe function. f(x)=x 6 4x

D: 3, R: ,D: , R: 13,

D: ,3 R: 13,

D: ,-3 R: -13,