3.5 Transformation of functions  (Page 6/21)

Try It

Given the toolkit function $f(x)=x^2,$ graph $g(x)=-f(x)$ and $h(x)=f(-x).$ Take note of any surprising behavior for these functions.

Notice: $g(x)=f(-x)$ looks the same as $f(x)$ .

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Determining even and odd functions

Some functions exhibit symmetry so that reflections result in the original graph. For example, horizontally reflecting the toolkit functions $f(x)=x^2$ or $f(x)=\left|x\right|$ will result in the original graph. We say that these types of graphs are symmetric about the y -axis. A function whose graph is symmetric about the y -axis is called an even function.

If the graphs of $f(x)=x^3$ or $f(x)=\frac{1}{x}$ were reflected over both axes, the result would be the original graph, as shown in [link] .

We say that these graphs are symmetric about the origin. A function with a graph that is symmetric about the origin is called an odd function .

Note: A function can be neither even nor odd if it does not exhibit either symmetry. For example, $f(x)=2^x$ is neither even nor odd. Also, the only function that is both even and odd is the constant function $f(x)=0.$

Graphing functions using stretches and compressions

Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.

We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.

Vertical stretches and compressions

When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch ; if the constant is between 0 and 1, we get a vertical compression . [link] shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.