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Analyzing Graphs of Functions and Relations

Analyzing Function Graphs

By looking at the graph of a function, you can determine the function's domain and range and estimate the x- and y-intercepts. The x-intercepts of the graph of a function are also called the zeros of the function because these input values give an output of 0.

Example Use the graph of f to find the domain and range of the function and to approximate the y-intercept and zero(s). Then confirm the estimate algebraically.

The graph is not bounded on the left or right, so the domain is the set of all real numbers.

                              {xx}

The graph does not extend above 5.0625 or f(-0.75), so the range is all real numbers less than or equal to 5.0625.

                              {yy ≤ 5.0625, y}

The y-intercept is the point where the graph intersects the y-axis. It appears to be 4.5. Likewise, the zeros are the x-coordinates of the points where the graph crosses the x-axis. They seem to occur at -3 and 1.5.

To find the y-intercept algebraically, find f(0).
                              f(0) = -(0)2 − 1.5(0) + 4.5 = 4.5

To find the zeros algebraically, let f(x) = 0 and solve for x.
                                   -x2 − 1.5x + 4.5 = 0
                                 -1(x + 3)(x − 1.5) = 0
                                           x = -3 or x = 1.5



Exercises

Use the graph of g to find the domain and range of the function and to approximate its y-intercept and zero(s). Then find its y-intercept and zeros algebraically.











Chapter 1 10 Glencoe Precalculus


Symmetry of Graphs

A graph of a relation that is symmetric to the x-axis and/or the y-axis has line symmetry. A graph of a relation that is symmetric to the origin has point symmetry.

Symmetric with respect to... Description Algebraic Test
x-axis For every (x, y) on the graph, (x, -y) is also on the graph. Replacing y with -y produces an equivalent equation.
y-axis For every (x, y) on the graph, (-x, y) is also on the graph. Replacing x with -x produces an equivalent equation.
origin For every (x, y) on the graph, (-x, -y) is also on the graph. Replacing x with -x and y with -y produces an equivalent equation.

Functions that are symmetric with respect to the y-axis are even functions, and for every x in the domain, f(-x) = f(x). Functions that are symmetric with respect to the origin are odd functions and for every x in the domain, f(-x) = -f(x).

Example GRAPHING CALCULATOR Graph f(x) = -x3 + 2x. Analyze the graph to determine whether the function is even, odd, or neither. Confirm algebraically. If odd or even, describe the symmetry of the graph of the function.

From the graph, it appears that the function is symmetric to the origin.
Confirm: f(-x) = -(-x)3 + 2(-x) = x3 − 2x = -f(x)
The function is odd because f(-x) = -f(x).









Exercises

GRAPHING CALCULATOR Graph each function. Analyze the graph to determine whether each function is even, odd, or neither. Confirm algebraically. If odd or even, describe the symmetry of the graph of the function.

  1. f(x) = 4x3 + 1




  2. g(x) = x4 − 10x2 + 9









  3. g(x) = x3 − 6x


Chapter 1 11 Glencoe Precalculus