Digital Lesson Shifting Graphs

Example: Shift, Reflection The graphs of many functions are transformations of the graphs of very basic functions. Example: The graph of y = x2 + 3 is the graph of y = x2 shifted upward three units. This is a vertical shift. x y -4 4 -8 8 y = x2 + 3 y = x2 The graph of y = -x2 is the reflection of the graph of y = x2 in the x-axis. y = -x2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Shift, Reflection

Vertical Shifts If c is a positive real number, the graph of f(x) + c is the graph of y = f(x) shifted upward c units. If c is a positive real number, the graph of f(x) – c is the graph of y = f(x) shifted downward c units. x y f(x) + c f(x) +c f(x) – c -c Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Vertical Shifts

Example: Vertical Shifts Example: Use the graph of f (x) = |x| to graph the functions f (x) = |x| + 3 and f (x) = |x| – 4. x y -4 4 8 f (x) = |x| + 3 f (x) = |x| f (x) = |x| – 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Vertical Shifts

Horizontal Shifts If c is a positive real number, then the graph of f (x – c) is the graph of y = f (x) shifted to the right c units. x y -c +c If c is a positive real number, then the graph of f (x + c) is the graph of y = f (x) shifted to the left c units. y = f (x + c) y = f (x) y = f (x – c) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Horizontal Shifts

Example: Horizontal Shifts Example: Use the graph of f (x) = x3 to graph f (x) = (x – 2)3 and f (x) = (x + 4)2 . x y -4 4 f(x) = x3 f (x) = (x + 4)2 f (x) = (x – 2)3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Horizontal Shifts

Example: Vertical and Horizontal Shifts Example: Graph the function using the graph of . First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. -4 x y 4 -4 y 4 x (4, 2) (0, 0) (-1, -2) (4, -2) (0, -4) (-5, -4) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Vertical and Horizontal Shifts

Reflection in the y-Axis and x-Axis. The graph of a function may be a reflection of the graph of a basic function. The graph of the function y = f (-x) is the graph of y = f (x) reflected in the y-axis. x y y = f (-x) y = f (x) The graph of the function y = -f (x) is the graph of y = f (x) reflected in the x-axis. y = -f (x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Reflection in the y-Axis and x-Axis.

Example: Graph y = -(x + 3)2 using the graph of y = x2. First reflect the graph in the x-axis. Then shift the graph three units to the left. x y -4 4 x y 4 y = x2 (-3, 0) y = - (x + 3)2 y = - x2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Reflections

Vertical Stretching and Shrinking If c > 1 then the graph of y = c f (x) is the graph of y = f (x) stretched vertically by c. If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x) shrunk vertically by c. - 4 x y 4 y = x2 y = 2x2 Example: y = 2x2 is the graph of y = x2 stretched vertically by 2. is the graph of y = x2 shrunk vertically by . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Vertical Stretching and Shrinking

Example: Vertical Stretch and Shrink Horizontal Stretching and Shrinking If c > 1, the graph of y = f (cx) is the graph of y = f (x) shrunk horizontally by c. If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x) stretched horizontally by c. y = |2x| Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2. - 4 x y 4 y = |x| is the graph of y = |x| stretched horizontally by . Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Vertical Stretch and Shrink

Example: Multiple Transformations Example: Graph given the graph y = x3. Graph y = x3 and do one transformation at a time. - 4 4 8 x y - 4 4 8 x y Step 3: Step 1: y = x3 Step 4: Step 2: y = (x + 1)3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Multiple Transformations