1.2 day 2 Transformations of Functions Northern California Coast, Pebble Beach, California Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2009

Pebble Beach, California Seal Rock Pebble Beach, California Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2006

The rules for shifting, stretching, shrinking, and reflecting the graph of a function make it easier to sketch functions by hand. Since we will be frequently using graphs in our study of calculus, we will do a quick review of those rules. If we know how to graph the “parent” graph of a function, then we can modify that graph to get the one we want.

Example: Adding a positive number at the end moves the graph up.

Example: Adding a constant to x inside the parentheses moves the graph to the left. The horizontal changes happen in the opposite direction to what you might expect.

Example: Placing a coefficient in front of the function causes a vertical stretch. In this case, the graph goes up twice as fast.

Example: If the coefficient is negative, then the graph is reflected about the y-axis.

Example: Placing a coefficient inside the function in front of the x causes a horizontal shrink. In this case, the graph expands horizontally half as fast. The horizontal changes happen in the opposite direction to what you might expect.

Example: Clearing the parentheses: In this case, a horizontal shrink is the same as a vertical stretch, but this is not always true.

Example: If the coefficient inside the function in front of the x is negative, you get a reflection about the y axis. In this case, since we started with an even function, we cannot see the reflection. Let’s look at an odd function.

Example: Placing a negative coefficient inside the function in front of the x causes a reflection about the y-axis.

To summarize the rules for transformations of graphs: Vertical stretch or shrink; reflection about x-axis Vertical shift Positive d moves up. is a stretch. Horizontal shift Horizontal stretch or shrink; reflection about y-axis Positive c moves left. is a shrink. The horizontal changes happen in the opposite direction to what you might expect.

p Now let’s look at a more complicated example: vertical right three flip right three moves down half as fast up four p