Precalculus 20151.4 Transformation of Functions Objectives

Recognize graphs of common functions

Use shifts to graph functions

Use reflections to graph functions

Use stretching & shrinking to graph functions

Graph functions w/ sequence of transformations

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

The identity function f(x) = x

The squaring function

2)( xxf

xxf )(

The square root function

xxf )(The absolute value function

3)( xxf

The cubing function

The cube root function 3( )f x x

Vertical and Horizontal Shifts

Let c be a positive real number. Vertical and horizontal shifts in the graph of y = f(c) are represented as follows:

1.Vertical shift c units upward:

2.Vertical shift c units downward:

3.Horizontal shift c units to the right:

4.Horizontal shift c units to the left:

h x f x c

h x f x c

h x f x c

h x f x c

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

Graph Illustrating Vertical Shift.

Vertical Translation

Vertical TranslationFor c > 0, the graph of y = f(x) + c is the graph of y = f(x) shifted up c units;

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

Graph Illustrating Horizontal Shift.

Horizontal TranslationFor c > 0, the graph of y = f(x c) is the graph of y = f(x) shifted right c units;the graph of y = f(x + c) is the graph of y = f(x) shifted left c units.

Why translations work the way they do

Upward Vertical TranslationConsider the function f(x) = x2 . If we add, say 4 units, to f(x) then the function becomes g(x) = f(x) + 4. The graph of g(x) is an upward translation of the graph of f(x) shifted vertically by 4 units.  The reason why the graph shifted upward is because 4 units have been added to every y-coordinate of the graph of f(x), and the y-coordinate of f(x) happens to be f(x) itself or x2.

Thus, adding 4 to x2 causes the y-coordinate of every ordered pair of f(x) to increase by 4.

Why translations work the way they do

Horizontal TranslationConsider the function f(x) = x2 . In order for a function to have its graph shifted n units to the right, then all we have to do is add n units to every x-coordinate of the function.

The x-coordinate of a graph of a function can be found by solving for x. So if our function is y = x2, then solving for x:

If we want the function y = x2 to have its graph shifted to the right, say 3 units, then we add 3 to the right side of the equation above as follows:

All the x-coordinates of f(x) have now been shifted 3 units to the right; and if we solve for y: 2

3y x

x y

3x y

3x y

Use the basic graph to sketch the following:

( ) 3f x x

2( ) 5f 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.

What is the equation of the translated function?

( ) , f x x

( ) 6 4 f x x

Reflections

The graph of f(x) is the reflection of the graph of f(x) across the x-axis.

The graph of f(x) is the reflection of the graph of f(x) across the y-axis.

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Digital Figures, 1–22

Graph of a Reflection across the x-axis.

What would f(x) look like if it were reflected across the y-axis?

Graph of a Reflection across the y-axis.

Use the basic graphs to sketch each of the following:

( )f x x

( )f x x 2( )f x x

( )f x x

Vertical Stretching and Shrinking

The graph of af(x) can be obtained from the graph of f(x) by

stretching vertically for |a| > 1, orshrinking vertically for 0 < |a| < 1.

For a < 0, the graph is also reflected across the x-axis.

VERTICAL STRETCH (SHRINK)y’s do what we

think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

2( ) 3 4f x x

2( ) 4f x x

21( ) 4

2f x x

Horizontal Stretching or Shrinking

The graph of y = f(cx) can be obtained from the graph of y = f(x) by

shrinking horizontally for |c| > 1, orstretching horizontally for 0 < |c| < 1.

For c < 0, the graph is also reflected across the y-axis.

Horizontal stretch & shrink

Think of the coefficient on x as speed. It will either speed up or slow down the function.

( ) sin(2 )g x x

( ) sinf x x

Transforming points

Use transformations to find 2 points on the graph of g(x)

Write the function g(x)

3( )

g(x) = 3 ( 2) 1

f x x

f x

3g(x)= 3( 2) 1x

0,0 2, 1

1,1 1,2

X values are shifted 2 units left. Y values (function values) are multiplied by 3 and shifted down 1 unit.

Graph of Example3

3

( )

( ) 3 ( 2) 1 3( 2) 1

f x x

g x f x x

(0,0), (1,1)

(-2,-1), (-1,2)

The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x).

g(x) = f(x-2)

g(x)= 4f(x)

g(x) = f(½x)

g(x) = -f(x)

(-10, 4)

(-12, 16)

(-24, 4)

(-12, -4)

Discuss with your neighborCompare the graph of the function

below with the graph of . What transformations have taken place from the basic graph?

3( )f x x

32 1 3h x x

•Shift right by 1 unit•Vertical stretch by a factor of 2•Reflect across x-axis•Vertical shift up by 3

Homework

Pg. 49: 3,9,11, 15-31 odd, 39-43 odd, 51, 59