Transformations of Graphs 3.5 Graph functions using vertical and horizontal shifts Graph functions using stretching and shrinking Graph functions using reflections Combine transformations Model data with transformations (optional)

Vertical and Horizontal Shifts We use these two graphs to demonstrate shifts, or translations, in the xy-plane.

Vertical Shifts A graph is shifted up or down. The shape of the graph is not changed—only its position. Every point moves upward 2.

Horizontal Shifts A graph is shifted right: replace x with (x – 2) Every point moves right 2.

Horizontal Shifts A graph is shifted left: replace x with (x + 3), Every point moves left 3.

Vertical and Horizontal Shifts Let f be a function, and let c be a positive number.

Combining Shifts Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally. Shift the graph of y = |x| to the right 2 units and downward 4 units. y = |x| y = |x – 2| y = |x – 2|  4

Example: Combining vertical and horizontal shifts Complete the following. (a) Write an equation that shifts the graph of f(x) = x2 left 2 units. Graph your equation. (b) Write an equation that shifts the graph of f(x) = x2 left 2 units and downward 3 units. Graph your equation. Solution To shift the graph left 2 units, replace x with x + 2.

Example: Combining vertical and horizontal shifts (b) Write an equation that shifts the graph of f(x) = x2 left 2 units and downward 3 units. Graph your equation. Solution To shift the graph left 2 units, and downward 3 units, we subtract 3 from the equation found in part (a).

Vertical Stretching and Shrinking If the point (x, y) lies on the graph of y = f(x), then the point (x, cy) lies on the graph of y = cf(x). If c > 1, the graph of y = cf(x) is a vertical stretching of the graph of y = f(x), whereas if 0 < c < 1 the graph of y = cf(x) is a vertical shrinking of the graph of y = f(x).

Vertical Stretching and Shrinking

Horizontal Stretching and Shrinking If the point (x, y) lies on the graph of y = f(x), then the point (x/c, y) lies on the graph of y = f(cx). If c > 1, the graph of y = f(cx) is a horizontal shrinking of the graph of y = f(x), whereas if 0 < c < 1 the graph of y = f(cx) is a horizontal stretching of the graph of y = f(x).

Horizontal Stretching and Shrinking

Example: Stretching and shrinking of a graph Use the graph of y = f(x) to sketch the graph of each equation. a) y = 3f(x) b)

Example: Stretching and shrinking of a graph Solution a) y = 3f(x) Vertical stretching Multiply each y-coordinate on the graph by 3. (1, –2  3) = (1, –6) (0, 1  3) = (0, 3) (2, –1  3) = (2, –3)

Example: Stretching and shrinking of a graph Solution continued b) Horizontal stretching Multiply each x-coordinate on the graph by 2 or divide by ½. (1  2, –2) = (2, –2) (0  2, 1) = (0, 1) (2  2, –1) = (4, –1)

Reflection of Graphs Across the x- and y-Axes 1. The graph of y = –f(x) is a reflection of the graph of y = f(x) across the x-axis. 2. The graph of y = f(–x) is a reflection of the graph of y = f(x) across the y-axis.

Reflection of Graphs Across the x- and y-axes

Example: Reflecting graphs of functions For the representation of f, graph the reflection across the x-axis and across the y-axis. The graph of f is a line graph determined by the table.

Example: Reflecting graphs of functions Solution Here’s the graph of y = f(x).

Example: Reflecting graphs of functions Solution continued To graph the reflection of f across the x-axis, start by making a table of values for y = –f(x) by negating each y-value in the table for f(x) .

Example: Reflecting graphs of functions Solution continued To graph the reflection of f across the y-axis, start by making a table of values for y = f(–x) by negating each x-value in the table for f(x) .

Combining Transformations Transformations of graphs can be combined to create new graphs. For example the graph of y = 2(x – 1)2 + 3 can be obtained by performing four transformations on the graph of y = x2.

Combining Transformations 1. Shift the graph 1 unit right: y = (x – 1)2. 2. Vertically stretch the graph by factor of 2: y = 2(x – 1)2. 3. Reflect the graph across the x-axis: y = 2(x – 1)2. 4. Shift the graph upward 3 units: y = 2(x – 1)2 + 3.

Combining Transformations continued Shift to the left 1 unit. Shift upward 3 units. Reflect across the x-axis. Stretch vertically by a factor of 2 y = 2(x – 1)2 + 3

Combining Transformations The graphs of the four transformations.

Combining Transformations The graphs of the four transformations.

Example: Combining transformations of graphs Describe how the graph of each equation can be obtained by transforming the graph of Then graph the equation.

Example: Combining transformations of graphs Solution Vertically shrink the graph by factor of 1/2 then reflect it across the x-axis.

Example: Combining transformations of graphs Solution continued Reflect it across the y-axis. Shift left 2 units. Shift down 1 unit.

Summary of Transformations

Combining Transformations Order: Horizontal transformation Stretching, shrinking and reflecting Vertical transformation Doing these in this order will protect the graph and ensure that you end up with the correct shape.