/ 1.4 Graphing Techniques: Transformations

Do you recall our Library of Functions from 1.4, which we also call “parent” functions?

Name / General equation
Constant function / f(x) = c
Identity/Linear function / f(x) = x
Squaring/quadratic function / f(x) = x2
Cubing/Cubic function / f(x) = x3
Square root function / f(x) =
Cube root function / f(x) =
Absolute value function / f(x) =
Greatest integer/step function / f(x) = [x]
Reciprocal/Rational function / f(x) =

If we are familiar with the graphs of these parent functions, we can make certain changes to the equation to produce related, “child” functions. These changes are called transformations and consist of:

Translations: graph slides either left/right or up/down

Reflections: graph turns over either the x-axis or y-axis

Dilations: graph either shrinks or stretches in size

Let’s look at each in turn…

Vertical and Horizontal Shifts (Translations)

Let c be a positive real number. Vertical and horizontal shifts in the graph of are represented:

Vertical shift upwardc units:

Vertical shift downwardc units:

Horizontal shift rightc units:

Horizontal shift left c units:

So, the graph of f(x) has the same shape and look of the graph of f(x) + c; it’s just moved vertically by “c” units.
Examples: Use horizontal and/or vertical shifts to sketch the following

Describe the shifts below each graph.

1. 2. 3.

Reflections in the Coordinate Axes

  • Reflection in the x-axis:
  • Reflection in the y-axis:

Examples: Sketch the graph of the following by using reflections.

Describe the shifts below each graph.

4. 5.

Nonrigid Transformations (Compressions and Stretches)

Vertical and horizontal shifts, and reflections are called rigid transformations because the basic shape of the graph is unchanged. Nonrigid transformations are those that cause a distortion – a change in the shape of the original graph.

A vertical stretch in if

A vertical shrink(compression) in if

Examples: Sketch the graphs of the following using stretching/shrinking transformations.

5. 6.

We can use this information to graph functions that contain a combination of transformations. In general, if we start with any parent function f(x) from our library, the function g(x) = a*f(bx-c) + dwill be:

Reflected over the x-axis
Compressed vertically (by factor of a)
Stretched vertically (by factor of a) / if a < 0
if 0< a < 1
if a > 1
Stretched horizontally (by factor of 1/b)
Compressed horizontally (by 1/b)
Reflected over the y-axis / if 0< b < 1
if b > 1
if b < 0
Shifted c units horizontally, to the left (-c/b units)
Shifted c units horizontally, to the right (-c/b units) / If c 0
If c 0
Shifted d units vertically, up
Shifted d units vertically, down / If d > 0
If d < 0

Graph the following using transformations of parent functions.

7. f(x) = 2(x + 2)3 – 3

Solution:

a)a = 2, b = 1, c = 2, and d = –3
b)Identify the parent function: ______
c)What effect will a = 2 have on the parent graph? ______
d)What effect will b = 1 have on the parent graph? ______
e)What effect will c = 2 have on the parent graph? ______
f)What effect will d = –3 have on the parent graph? ______
g)Now sketch the graph, noting these effects. Plot vertex and plug in values of x to find 2 other points. Use parent as a guide for general shape.
8. Now plot this: f(x) =
a = ______; b = ______;
c = ______; d = ______/

Math Analysis; updated 2/9/2011Page 1