In Exercises 19–24, (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. $f\left(x\right)=4x-x^3$

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1. Identify the degree and leading coefficient of the polynomial function \(f(x) = 4x - x^3\). The highest power of \(x\) is 3, so the degree is 3, and the leading coefficient is the coefficient of \(x^3\), which is \(-1\).
2. Apply the Leading Coefficient Test to determine the end behavior: Since the degree is odd (3) and the leading coefficient is negative (\(-1\)), as \(x \to \infty\), \(f(x) \to -\infty\), and as \(x \to -\infty\), \(f(x) \to \infty\).
3. Check for symmetry by testing \(f(-x)\) and comparing it to \(f(x)\) and \(-f(x)\). Calculate \(f(-x) = 4(-x) - (-x)^3 = -4x + x^3\). Since \(f(-x) \neq f(x)\) and \(f(-x) \neq -f(x)\), the function has neither y-axis symmetry nor origin symmetry.
4. To graph the function, start by plotting key points such as the y-intercept at \(x=0\), where \(f(0) = 0\), and other points by substituting values of \(x\) (e.g., \(x=1, -1, 2, -2\)) to understand the shape of the curve.
5. Use the end behavior from step 2 and the points from step 4 to sketch the graph, showing that the graph falls to the right and rises to the left, consistent with the Leading Coefficient Test for an odd degree polynomial with a negative leading coefficient.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps determine the end behavior of a polynomial function by examining the degree and leading coefficient. For odd-degree polynomials, the ends of the graph go in opposite directions, while for even-degree polynomials, both ends go the same way. The sign of the leading coefficient indicates whether the graph rises or falls on the right end.

Symmetry of Functions

Symmetry in graphs refers to whether a function is even, odd, or neither. Even functions have y-axis symmetry, meaning f(x) = f(-x). Odd functions have origin symmetry, meaning f(-x) = -f(x). Checking these properties helps understand the shape and behavior of the graph.

Graphing Polynomial Functions

Graphing polynomials involves plotting key points, analyzing end behavior, and identifying intercepts and symmetry. Understanding the degree and coefficients guides the shape, while evaluating the function at specific x-values helps plot accurate points for a complete graph.