This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn:

- how to graph some basic functions
- how to graph piece-wise defined functions
- reflection of graphs in the
*x*-axis or*y-*axis - horizontal and vertical graph transformations
- vertical stretching and shrinking graphs
- graphs of inverse functions
- how to find the inverse function using algebra

The coordinate plane can be used for graphing functions. To graph a function in the *xy*-plane, we
represent each input *x* and its corresponding output *f*(*x*) as a point (*x, y*), where *y* = *f*(*x*). In other words, you use the *x*-axis for the input and the *y*-axis for the output.

The following video shows how to sketch the graph of six basic functions: *f*(*x*) = *x*, *f*(*x*) = *x*^{2}, *f*(*x*) = *x*^{3}, *f*(*x*) = square root of *x*, *f*(*x*) = cube root of *x*, *f*(*x*) = absolute value of *x*

A **piecewise-defined function** (also called a **piecewise function**) is a function whose definition changes depending on the value of the input variable.

This video shows how to evaluate and graph piece-wise functions.

If the graph of *y* = *f*(*x*) is known, the

*h*(*x*) = −*f*(*x*) is a reflection in the*x*-axis*h*(*x*) =*f*(*− x*) is a reflection in the*y*-axis

In general, for any function *h*(*x*) and any positive number *c*, the following are true.

• The graph of *h*(*x*) + c is the graph of *h*(*x*) shifted upward by *c* units.

• The graph of *h*(*x*) − c is the graph of *h*(*x*) shifted downward by *c* units.

• The graph of *h*(*x* + c) is the graph of *h*(*x*) shifted to the left by *c* units.

• The graph of *h*(*x* − c) is the graph of *h*(*x*) shifted to the right by *c* units.

This video gives 9 examples to illustrate the basic outline of doing horizontal and vertical translations of graphs.

In general, for any function *h*(*x*) and any positive number *c*, the following are true.

- The graph of
*ch*(*x*) is the graph of*h*(*x*) stretched vertically by a factor of*c*if*c*> 1. - The graph of
*ch*(*x*) is the graph of*h*(*x*) shrunk vertically by a factor of*c*if 0 < c < 1.

This video shows how multiplying by a number will affect the graph. Vertical stretches and compressions.

This video shows how to graph an inverse function and points out that a graph of a function and its inverse is symmetrical about the line *y = x*.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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