Problem 52 - Algebra and Trigonometry

Question

a. Use the Leading Coefficient Test to determine the graph's end behavior. b. Find the \(x\)-intercepts. State whether the graph crosses the \(x\)-axis, or touches the \(x\) -axis and turns around, at each intercept. c. Find the \(y\)-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. \(f(x)=6 x-x^{3}-x^{5}\)

Step-by-Step Solution

Answer

The graph of \(f(x) = 6x - x^3 - x^5\) falls to the left and right, crosses the \(x\)-axis at \(0, ±1, ±\sqrt{6}\), and intersects the \(y\)-axis at \(0\). The graph has no symmetry and may have up to 4 turning points.

Step 1: End Behavior

First, we will find the end behavior of the graph using the Leading Coefficient Test. The highest power in the polynomial \(f(x) = 6x - x^3 - x^5\) is 5, and the leading term of the polynomial is \(-x^5\). Since the power is odd and the leading coefficient is negative, the end behavior is that the graph of \(f(x)\) falls to the left and falls to the right.

Step 2: \(x\)-intercepts and Behavior

Now, find the \(x\)-intercepts by setting \(f(x) = 0\) and solving the equation. With the polynomial \(f(x) = 6x - x^3 - x^5\), it factors into \(x(6 - x^2)(1 - x^2)\). Setting this equal to 0, we find the \(x\)-intercepts as \(x = 0\), \(x = ±\sqrt{6}\) and \(x = ±1\). The graph will cross the \(x\)-axis at these intercepts, as shown by the fact that the multiplicity of each zero is 1, an odd number.

Step 3: \(y\)-intercept

The \(y\)-intercept is found by evaluating the function at \(x = 0\). In the equation \(f(x) = 6x - x^3 -x^5\), when \(x = 0\), \(f(0) = 0\), so the \(y\)-intercept is \(y = 0\).

Step 4: Symmetry

To determine symmetry, replace \(x\) with \(-x\) in the equation. Doing so gives \(f(-x)=-6x +x^{3}+ x^{5}\), which is not the same as \(f(x)\) nor is it the negation of \(f(x)\). Therefore, the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Graphing

Without specific values to graph, simply use the results from the above steps. However, the maximum number of turning points a polynomial can have is one less than its degree, so the graph of this function can have at most 4 turning points as a check.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a crucial tool to predict a polynomial function's end behavior. For the function \( f(x) = 6x - x^3 - x^5 \), the leading term is \(-x^5\). Here are some key observations:

• The degree of the polynomial is 5, which is an odd number.
• The leading coefficient is -1, a negative number.

This information tells us that as \( x \) moves toward positive infinity, \( f(x) \) will fall, and as \( x \) moves toward negative infinity, \( f(x) \) will also fall. This is because odd degree polynomials behave differently at each end, and a negative leading coefficient flips the normal behavior, resulting in both ends moving downward.

End Behavior

Understanding the end behavior of polynomial functions helps us visualize their long-term trends. For the polynomial \( f(x) = 6x - x^3 - x^5 \), we already know from the Leading Coefficient Test that the graph falls on both ends.- **As \( x \rightarrow +\infty \):** Since the leading coefficient is negative and the degree is odd, \( f(x) \rightarrow -\infty \). The graph heads downward.- **As \( x \rightarrow -\infty \):** Similarly, \( f(x) \rightarrow -\infty \), and the graph continues to descend.This behavior helps us set the general direction of our graph even before pinpointing other critical aspects like intercepts.

x-intercepts

To find the \( x \)-intercepts, set the polynomial equal to zero and solve: \( f(x) = 0 \). In the equation \( f(x) = 6x - x^3 - x^5 \), it factors to \( x(6 - x^2)(1 - x^2) = 0 \). Thus, the \( x \)-intercepts are:

• \( x = 0 \)
• \( x = \pm \sqrt{6} \)
• \( x = \pm 1 \)

Each of these intercepts has a multiplicity of 1 (they occur only once), indicating that the graph of \( f(x) \) crosses the \( x \)-axis at each point. These intercepts provide critical points for sketching the polynomial graph accurately.

y-intercept

Finding the \( y \)-intercept of a polynomial involves evaluating the function at \( x = 0 \). For our function, \( f(x) = 6x - x^3 - x^5 \), this simply becomes:\[ f(0) = 6 \times 0 - 0^3 - 0^5 = 0 \]Therefore, the \( y \)-intercept is \( y = 0 \). This means that the graph passes through the origin (0,0), which is an important point where the function intersects the \( y \)-axis.

Graph Symmetry

Examining a function's symmetry can simplify graphing and understanding it. For \( f(x) = 6x - x^3 - x^5 \), checking for symmetry involves substituting \( x \) with \(-x\) and comparing with \( f(x) \):\[ f(-x) = -6x + x^3 + x^5 \]Since \( f(-x) \) is neither equal to \( f(x) \) nor \(-f(x)\), the graph lacks both y-axis and origin symmetry. Thus, the function doesn't exhibit symmetric patterns, which informs us that the graph might veer uniquely in different quadrants without mirroring itself across an axis or the origin.