8.6 1

Section 8.6 Quadratic Functions & Graphs

Graphing Parabolas f(x)=ax2

f(x)=ax2+k f(x)=a(x–h)2

f(x)=a(x–h)2+k Finding the Vertex and Axis of Symmetry Finding Maximums or Minimums

8.6 2

Definition

Let’s make a table of values for f(x) = x2

Then sketch the basic parabola shape …

8.6 3

Make a table of values for f(x) = x2

Then graph it

8.6 4

Similar Curves Changing the

coefficient of x2: Smaller is wider Larger is narrower

The vertex remains the same

8.6 5

Make a table of values for f(x) = -½x2

Then graph it

8.6 6

Reflections The graph of y= -ƒ(x) is the graph of y = ƒ(x)

reflected about the x-axis.

)(xf

8.6 7

Vertical Translations If ƒ is a function and k is a positive number, then The graph of y = ƒ(x) + k is identical to the

graph of y =ƒ(x) except that it is translated k units upward.

The graph of y = ƒ(x) - k is identical to the graph of y = ƒ(x) except that it is translated k units downward.

Sketch f(x) = x2 + 3 on the board

kxf )(

8.6 8

Horizontal TranslationsIf ƒ is a function and h is a positive number, Then the graph of y = ƒ(x - h) is identical to

the graph of y = ƒ(x) except that it is translated h units to the right.

The graph of y = ƒ(x + h) is identical to the graph of y = ƒ(x) except that it is translated h units to the left.

)( hxf

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Graph f(x) = (x – 3)2

8.6 10

Graph g(x) = -2(x +4)2

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khxaxf 2)()(Equation The

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Graph g(x) = (x – 3)2 – 5

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Finding the Vertex and Axis of SymmetryAnswer the following questions about the given

equation:

a. Does the graph open upward or downward?

b. What are the coordinates of the vertex?

c. What is the axis of symmetry?

d. Is it narrow or wide?

Up

(5,1)

x=5narrow

156 2 xy

8.6 14

Finding the Vertex and Axis of SymmetryAnswer the following questions about the given

equation:

a. Does the graph open upward or downward?

b. What are the coordinates of the vertex?

c. What is the axis of symmetry?

d. Is it narrow or wide?

Down

(-4,-3)

x=-4wide

34 2

32 xy

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Maximums & Minimums

8.6 16

Maximum or Minimum?Vertex and Axis of Symmetry?

4)0,4(0)4()(

3)5,3(55)3(2

2)6,2(66)2()(

7)5,7(55)7()(

232

2

221

2

xAxisyMaxxxf

xAxisyMaxxy

xAxisyMinxxh

xAxisyMinxxg

8.6 17

But what if the function is not in the form of f(x) = a(x – h)2 + k ?

5)3()(

3436)(

46)(

46)(

!

2

222

2

2

xxf

xxxf

xxxf

xxxf

squaretheComplete

1)2(3)(

2313)24(3)(

13)4(3)(

13123)(

!

2

222

2

2

xxf

xxxf

xxxf

xxxf

squaretheComplete

8.6 18

What Next? Section 9.1Composite Functions

Sections 8.7 - 8.9 may not be covered