Graphs of Functions

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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

Graphs of Functions

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) = x², f(x) = x³, f(x) = square root of x, f(x) = cube root of x, f(x) = absolute value of x

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Piecewise Functions

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.

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This video gives 2 more examples of Graphing Piecewise Defined Functions

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Reflection of Graphs

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

The following video shows how to reflect graphs over the x-axis and the y-axis

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Horizontal and Vertical Graph Transformations

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.

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This video shows some examples of horizontal and vertical translations of graphs.

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Vertically Stretching and Shrinking 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.

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This video shows how to vertically stretch and shrink graphs of functions.

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Graphs of Inverse Functions

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.

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How to find the inverse function using algebra

These videos give examples of finding the inverse of a function

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