precalculus 2015 1.4 transformation of functions objectives recognize graphs of common functions use...

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.

Copyright © Houghton Mifflin Company. All rights reserved.

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