Precalculus 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

The square root function

The absolute value function

The cubing function

The cube root function

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: 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 For 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.

Vertical Translation For 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.

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–13 Graph Illustrating Vertical Shift.

Graph Illustrating Horizontal Shift.

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 1–15 Graph Illustrating Horizontal Shift.

Horizontal Translation For 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 Translation Consider the function f(x) = x 2. 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 x 2. Thus, adding 4 to x 2 causes the y-coordinate of every ordered pair of f(x) to increase by 4.

Why translations work the way they do Horizontal Translation Consider the function f(x) = x 2. 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 = x 2, then solving for x: If we want the function y = x 2 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:

Use the basic graph to sketch the following:

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?

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:

Vertical Stretching and Shrinking The graph of af(x) can be obtained from the graph of f(x) by stretching vertically for |a| > 1, or shrinking 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!)

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, or stretching 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.

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

Transforming points Use transformations to find 2 points on the graph of g(x) Write the function g(x) X values are shifted 2 units left. Y values (function values) are multiplied by 3 and shifted down 1 unit.

Graph of Example (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 neighbor Compare the graph of the function below with the graph of. What transformations have taken place from the basic graph? 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, odd, odd, 51, 59