In Exercises 33–38, a. Use the Leading Coefficient Test to...

In Exercises 33–38,
a. Use the Leading Coefficient Test to determine the graph’s end behavior.
b. Determine whether the graph has y@axis symmetry, origin symmetry, or neither.
c. Graph the function.
$$
f(x)=-x^{4}+25 x^{2}
$$

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Key Concepts

Leading Coefficient Test

This is a method used to determine the end behavior of a polynomial function. By focusing on the sign and degree of the leading term, one can predict whether the graph’s arms will rise or fall as x approaches positive or negative infinity.

End Behavior of Polynomial Functions

End behavior describes how the values of a polynomial function behave as x approaches infinity or negative infinity. It is dictated primarily by the term of highest degree in the polynomial, which dominates the function's behavior for large absolute values of x.

Function Symmetry

Function symmetry involves determining whether a graph remains unchanged when reflected about the y-axis or rotated about the origin. A function is even if f(-x) equals f(x), indicating y-axis symmetry, and odd if f(-x) equals -f(x), indicating origin symmetry. This concept helps simplify the analysis and graphing process.

Graphing Polynomial Functions

Graphing a polynomial function involves identifying key features such as intercepts, turning points, and regions where the function increases or decreases. It often requires finding the zeros of the function and applying tests for end behavior and symmetry to create an accurate sketch of the function’s overall shape.

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