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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. |
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 |
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) = -x^{3} + 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. |
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.
Chapter 1 | 11 | Glencoe Precalculus |