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.
$$
f(x)=-x^{4}+6 x^{3}-9 x^{2}
$$

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Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of a polynomial function by considering the sign and the degree of the leading term. Specifically, when the degree is even or odd and whether the leading coefficient is positive or negative, it indicates how the function behaves as x approaches infinity or negative infinity.

Symmetry of Functions

Symmetry tests are used to determine if a function is symmetric with respect to the y-axis or the origin. A function is y-axis symmetric if replacing x with -x gives the same function, and it is origin symmetric if replacing both x and y with their negatives yields the same equation. These tests help in simplifying and understanding the graph of the function.

Graphing Polynomial Functions

Graphing polynomial functions involves analyzing critical aspects such as end behavior, intercepts, zeros, and turning points. By using techniques like factoring and applying symmetry and the Leading Coefficient Test, one can sketch an accurate graph of the function and understand the general shape and behavior of its graph.

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