1. 1 PRECALCULUS Section 1.6 Graphical Transformations

2. 2 The graphs of many functions are transformations of the graphs of very basic functions. The graph of y = –x 2 is the reflection of the graph of y = x 2 in the x-axis. Example: The graph of y = x 2 + 3 is the graph of y = x 2 shifted upward three units. This is a vertical shift. x y -4 4 4 -8 8 y = –x 2 y = x 2 + 3 y = x 2 Example: Shift, Reflection

3. Rigid Transformations leave the size and shape of a graph unchanged. It includes horizontal and vertical translations, reflections, or any combination of these. Non-rigid Transformations generally distort the shape of the graph. This includes horizontal and vertical stretches and/or shrinks.

4. For a positive real number c, vertical shifts of y = f(x) are: 1. Vertical shift c units upward: h(x) = y + c = f(x) + c 2. Vertical shift c units downward: h(x) = y  c = f(x)  c Vertical Shifts Vertical Shifts (rigid transformation)

5. h(x) = |x| – 2 Example : Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 2. f (x) = |x| x y -4 4 4 8 g(x) = |x| + 3 Example: Vertical Shifts

6. For a positive real number c, horizontal shifts of y = f(x) are: 1. Horizontal shift c units to right: h(x) = f(x  c) ; x  c = 0, x = c 2. Vertical shift c units to left: h(x) = f(x  c) ; x + c = 0, x = -c Horizontal Shifts Horizontal Shifts (rigid transformation)

7. f (x) = x 3 h(x) = (x + 4) 3 Example: Use the graph of f (x) = x 3 to graph g (x) = (x – 2) 3 and h(x) = (x + 4) 3. x y -4 4 4 g(x) = (x – 2) 3 Example: Horizontal Shifts

8. YOU TRY Describe how the graph y = | x | can be transformed to the graph of the given equation below. Sketch the graph. a. y = | x | - 4 b. y = | x + 2 |

9. y = | x | - 4 y = | x + 2 |

10. -4 y 4 x x y 4 Example: Graph the function using the graph of. First make a vertical shift 4 units downward. Then a horizontal shift 5 units left. (0, 0) (4, 2) (0, – 4) (4, –2) (– 5, –4) Example: Vertical and Horizontal Shifts (–1, –2)

11. EXAMPLE Below shows the graphs of y 1 = x 3 and a vertical or horizontal translation (y 2 ). Write an equation for y 2 for each graph.

12. Reflections in the coordinate axes of the graph of y = f(x) are represented as follows. 1. Reflection in the x-axis: y =  f(x) (symmetric to x-axis): (x, y) and (x, -y) 2. Reflection in the y-axis: y = f(  x) (symmetric to y-axis): (x, y) and (-x, y) Reflections in the Axes

13. y = f (–x) y = f (x) y = –f (x) 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. The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis. x y Reflection in the y-Axis and x-Axis.

14. x y 4 4 y = x 2 y = – (x + 3) 2 Example: Graph y = –(x + 3) 2 using the graph of y = x 2. First reflect the graph in the x-axis. Then shift the graph three units to the left. x y – 4 4 4 -4 y = – x 2 (–3, 0) Example: Reflections

15. Example Find an equation for the reflection of across each axis

16. Stretches and Shrinks Stretches and Shrinks (non rigid transformations) Let c be a positive real number. Then the following transformations result in stretches and shrinks of the graph of y = f(x) HORIZONTAL STRETCHES or SHRINKS stretch by a factor of c if c > 1 shrink by a factor of c is c < 1 VERTICAL STRETCHES or SHRINKS y = c f(x) stretch by a factor of c if c > 1 shrink by a factor of c if c < 1

17. EXAMPLE Let C 1 be the curve defined by y 1 = f(x) = x 3 – 16x. Find equations for the following non-rigid transformations of C 1 : a. C 2 is a vertical stretch of C 1 by a factor of 3 C 2 = y 2 = 3 f(x) = 3(x 3 – 16x) = 3x 3 – 48x b.C 3 is a horizontal shrink of C 1 by a factor of ½ C 3 = y 3 = = f(2x) = (2x) 3 – 16(2x) = 8x 3 – 32x

18. COMBINE FUNCTIONS The graph of y = x 2 undergoes the following transformations, in order. Find the equation of the graph that results. *a horizontal shift 2 units to the right y = (x – 2) 2 *a vertical stretch by a factor of 3 y = 3(x – 2) 2 *a vertical translation 5 units up y = 3(x – 2) 2 + 5 = 3x 2 – 12x + 17

19. EXAMPLE The graph of f(x) is shown. Determine the graph of y = 2f(x + 1) – 3 by showing the transformations of the graph f(x).