Unit 3 Functions (Linear and

Exponentials)

Parent Functions and Transformations

Transformation of FunctionsRecognize graphs of common functions

Use shifts to graph functionsUse reflections to graph functionsGraph functions w/ sequence of transformations

The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

The identity function f(x) = x

The exponential function ( ) xf x b

The quadratic function

2)( xxf

xxf )(

The square root function

xxf )(The absolute value function

3)( xxf

The cubic function

The rational function1

( )f xx

Transformations happen in 3 forms: (1) translations(2) reflections(3) stretching.

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function.

Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

( )y f x h k

Vertical Translation

OUTSIDE IS TRUE!Vertical Translationthe graph of y = f(x) + k is the graph of y = f(x) shifted up k units;

the graph of y = f(x) k is the graph of y = f(x) shifted down k units.

2( )f x x 2( ) 3f x x

2( ) 2f x x

Horizontal Translation

INSIDE LIES!Horizontal Translationthe graph of y = f(x h) is the graph of y = f(x) shifted right h units;

the graph of y = f(x + h) is the graph of y = f(x) shifted left h units.

2( )f x x

22y x 2

2y x

Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up

Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

3)(,)( 22 xxhxxf

22 )3()(,)( xxgxxf

2x

ExampleExplain the difference in the graphs

2( 3)y x 2 3y x

Horizontal Shift Left 3 Units

Vertical Shift Up 3 Units

Use the basic graph to sketch the following:

( ) 3f x x 2( ) 5f x x 3( ) ( 2)f x x ( ) 3f x x

Combining a vertical & horizontal shift

Example of function that is shifted down 4 units and right 6 units from the original function.

( )f x x

Use the basic graph to sketch the following:

( )f x x( )f x x 2( )f x x( )f x x

Reflection about the x-AxisThe graph of y = - f (x) is the graph of y

= f (x) reflected about the x-axis.

Reflection about the y-Axis• The graph of y = f (-x) is the graph of y =

f (x) reflected about the y-axis.

10

9

8

7

6

5

4

3

2

1

1 2 3 4-4 -3 -2 -1

f (x) = x2

h(x) =1/2x2

g(x) = 2x2

Stretching and Shrinking Graphs

Let f be a function and c a positive real number.•If a > 1, the graph of y = a f (x) is y = f (x) vertically stretched by multiplying each of its y-coordinates by a.

•If 0 < a < 1, the graph of y = a f (x) is y = f (x) vertically shrunk by multiplying each of its y-coordinates by a.

The big picture…

Sequence of Transformations

A function involving more than one transformation can be graphed by performing transformations in the following order.

1. Horizontal shifting (Parentheses)2. Vertical stretching or shrinking (Multiply)3. Reflecting (Multiply)4. Vertical shifting (Add/Subtract)

ExampleUse the graph of f(x) = x3 to graph g(x) = -

2(x+3)3 - 4

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

A combinationIf the parent function is

Describe the graph of

2y x

2( 3) 6y x The parent would be horizontally shifted right 3 units and vertically shifted up 6 units

If the parent function is

What do we know about

3y x32 5y x

The graph would be vertically shifted down 5 units and vertically stretched two times as much.

What can we tell about this graph?

3(2 )y xIt would be a cubic function reflected across the x-axis and horizontally compressed by a factor of ½.

Transformations of Exponential Functions

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Transformations of Graphs of Exponential Functions

Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.

State the domain, range and the horizontal asymptote for each.

( ) 2xf x

( ) 2 5xf x

30

Transformations of Graphs of Exponential Functions

Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.

State the domain, range and the horizontal asymptote for each.

( ) 2xf x

( ) 2 4xf x

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Transformations of Graphs of Exponential Functions

Describe the transformation(s) that the graph of must undergo in order to obtain the graph of each of the following functions.

State the domain, range and the horizontal asymptote for each.

( ) 2xf x

3( ) 2xf x

A) B)

C) D)

Graph using transformations and determine the domain, range and horizontal asymptote.