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

Vertical and Horizontal Shifts (1 of 2) We use these two graphs to demonstrate shifts, or translations, in the xy-plane. x −2 −1 1 2 y = x² 4

Vertical Shifts A graph is shifted up or down. The shape of the graph is not changed—only its position. x −2 −1 1 2 y = x² + 2 6 3

Horizontal Shifts (1 of 2) A graph is shifted right: replace x with (x − 2). x 1 2 3 4 y = (x − 2)²

Horizontal Shifts (2 of 2) A graph is shifted left: replace x with (x + 3). x −5 −4 −3 −2 −1 y = (x + 3)² 4 1

Vertical and Horizontal Shifts (2 of 2) Let f be a function, and let c be a positive number. To Graph Shift the Graph of y = f(x) by c Units y = f(x) + c upward y = f(x) − c downward y = f(x − c) right y = f(x + c) left

Combining Shifts Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally.

Example: Combining vertical and horizontal shifts (1 of 2) 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 a. To shift the graph left 2 units, replace x with x + 2. y = f(x + 2) or y = (x + 2)²

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

Vertical Stretching and Shrinking (1 of 2) 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 (2 of 2) x 1 4 f(x) 2 x 1 4 2f(x) 2 x 1 4 ½ f(x) ½

Horizontal Stretching and Shrinking (1 of 2) 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 (2 of 2) x −2 −1 1 2 f(x) 3 −3

Example: Stretching and shrinking of a graph (1 of 3) Use the graph of y = f(x) to sketch the graph of each equation. a. y = 3f(x) x −1 2 f(x) −2 1

Example: Stretching and shrinking of a graph (2 of 3) −1 2 3f(x) −6 3 −3 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 (3 of 3)

Reflection of Graphs Across the x- and y-axes (1 of 2)

Reflection of Graphs Across the x- and y-axes (2 of 2) 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.

Example: Reflecting graphs of functions (1 of 4) 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. x −2 −1 3 f(x) 1 −3 2

Example: Reflecting graphs of functions (2 of 4) Here’s the graph of y = f(x). x −2 −1 3 f(x) 1 −3 2

Example: Reflecting graphs of functions (3 of 4) 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). x −2 −1 3 −f(x) 1

Example: Reflecting graphs of functions (4 of 4) 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). Then plot these points and draw a line graph. x 2 1 −3 f(−x) −1

Combining Transformations (1 of 5) 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 (2 of 5) Shift the graph 1 unit right: y = (x − 1)² Vertically stretch the graph by factor of 2: y = 2(x − 1)² Reflect the graph across the x-axis: y = −2(x − 1)² Shift the graph upward 3 units: y = −2(x − 1)² + 3

Combining Transformations (3 of 5)

Combining Transformations (4 of 5) The graphs of the four transformations.

Combining Transformations (5 of 5) The graphs of the four transformations.

Example: Combining transformations of graphs (1 of 3)

Example: Combining transformations of graphs (2 of 3)

Example: Combining transformations of graphs (3 of 3) Shift right 2 units Reflect across the y-axis Shift down 1 unit

Modeling Data with Transformations Transformations of the graph of y = x² can be used to model some types of nonlinear data. By shifting, stretching, and shrinking this graph, we can transform it into a portion of a parabola that has the desired shape and location. In the next example we demonstrate this technique by modeling numbers of Walmart employees.

Example: Modeling data with a quadratic function (1 of 7) The table lists numbers of Walmart employees in millions for selected years. Walmart Employees (in millions) Year Employees 1987 0.20 1992 0.37 1997 0.68 2002 1.4 2007 2.2 Source: Walmart

Example: Modeling data with a quadratic function (2 of 7) a. Make a scatterplot of the data. b. Use transformations of graphs to determine f(x) =a(x − h)² + k so that f(x) models the data. Graph y = f(x) together with the data. c. Use f(x) to estimate the number of Walmart employees in 2015. Compare it with the actual value of 2.2 million employees.

Example: Modeling data with a quadratic function (3 of 7) Solution a. Here’s a calculator display of a scatterplot of the data.

Example: Modeling data with a quadratic function (4 of 7) b. It’s a parabola opening upward so a > 0. Vertex (minimum number of employees) could be (1987, 0.20): translate graph right 1987 units and up 0.20 unit. f(x) = a(x − 1987) + 0.20

Example: Modeling data with a quadratic function (5 of 7) To determine a, graph the data and y = f(x) for different values of a.

Example: Modeling data with a quadratic function (6 of 7) Experimenting yields a value of a near 0.005. So f(x) = 0.005(x − 1987) + 0.2.

Example: Modeling data with a quadratic function (7 of 7) c. To estimate the number of employees in 2015, evaluate ƒ(2015). ƒ(2015) = 0.005(2015 − 1987)² + 0.2 = 4.12 This model provides an estimate of about 4.12 million Walmart employees in 2015. The calculation involves extrapolation and is not accurate. It is almost double the actual value of 2.2 million.