Problem 45 - College Algebra

Question

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) 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 fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=6 x^{3}-9 x-x^{5}$$

Step-by-Step Solution

Answer

End behavior: as \(x \rightarrow \pm \infty\), \(f(x) \rightarrow -\infty\). x-intercepts are obtained from solutions to \(x(6x^{2} - 9 - x^{4}) = 0\). y-intercept is at (0,0). The function is neither symmetric about the y-axis nor the origin. The graph can be accurately drawn with the obtained points and expected turning points.

Step 1: End Behavior

Starting with the leading term, which is \(-x^{5}\), we can note that the degree of the polynomial (5) is odd and the leading coefficient (-1) is negative. Using the Leading Coefficient Test, this tells us that as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\) and that as \(x \rightarrow \infty\), \(f(x) \rightarrow -\infty\).

Step 2: Finding x-intercepts

Next, solve \(f(x) = 0\) which gives us, \(6x^{3} - 9x - x^{5} =0\). Factoring out an x, we get \(x(6x^{2} - 9 - x^{4}) = 0\). Solving this equation, we get \(x = 0\) as a solution and the quartic equation \(6x^{2} - 9 - x^{4} = 0\) for other solutions. Therefore, the x-intercepts are the solutions of the given equations. Depending on the roots' multiplicity, the graph either crosses the x-axis (for odd multiplicity) or touches the x-axis and turns around (for even multiplicity).

Step 3: Finding y-intercept

To find the y-intercept, set \(x = 0\) in \(f(x)\). On substitution, we get \(f(0) = -0 - 0 - 0 = 0\). Therefore, the y-intercept is (0,0).

Step 4: Symmetry Test

To check for symmetry with respect to y-axis, if \(f(-x) = f(x)\), then it is symmetric about the y-axis. If \(f(-x) = -f(x)\), the function is symmetric about the origin. Substituting \(-x\) in place of \(x\) in the given function, we get \(-f(x) \neq f(x)\) and \(f(-x) \neq f(x)\). Hence, the graph is neither symmetric with respect to the y-axis nor the origin.

Step 5: Graphing the function

The graph can be plotted with the above points as well as additional points found by substituting different x values. The degree of the polynomial (5) allows us to expect at most 4 ('n-1') turning points in the graph.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a useful tool to determine how the ends of a polynomial function's graph behave as the x-values approach positive and negative infinity. The leading term of a polynomial function is the term with the highest power of x. This term influences the end behavior of the polynomial. In this exercise, the leading term is \(-x^5\).

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

According to the Leading Coefficient Test:

• When the degree is odd and the leading coefficient is negative, as \(x \rightarrow \infty\), the function \(f(x) \rightarrow -\infty\).
• Similarly, as \(x \rightarrow -\infty\), \(f(x) \rightarrow \infty\).

These behaviors result in the graph falling to the left and to the right when visualized.

x-intercepts

Finding the x-intercepts involves setting the function equal to zero and solving for x. These are the points where the graph crosses or touches the x-axis. For our function \(f(x) = 6x^3 - 9x - x^5\), we set \(f(x) = 0\) to find:

• The equation becomes \(6x^3 - 9x - x^5 = 0\).
• Factoring out an x, we have \(x(6x^2 - 9 - x^4) = 0\).
• This gives us \(x = 0\) as one solution.
• The other solutions come from the quartic equation \(6x^2 - 9 - x^4 = 0\).

Each x-intercept found plays a unique role in how the graph interacts with the x-axis:

• If a root has an odd multiplicity, the graph crosses the x-axis at this point.
• If the multiplicity is even, the graph will touch the x-axis and turn around at this intercept.

y-intercept

To find the y-intercept, set \(x = 0\) in the polynomial function. This gives the point where the graph intersects the y-axis. For our function, substituting \(x = 0\) yields:

• \(f(0) = 6(0)^3 - 9(0) - (0)^5 = 0\).

Thus, the y-intercept is at \( (0, 0) \). This point is simple because all terms in the polynomial vanish when \(x = 0\). Knowing the y-intercept helps in plotting the graph accurately.

Symmetry

Symmetry in polynomial functions refers to the relation of the graph to the axes or the origin. Testing for symmetry can simplify graphing the function. To check for each type of symmetry, consider substituting \(-x\) for \(x\) in \(f(x)\):

• If \(f(-x) = f(x)\), the function has y-axis symmetry.
• If \(f(-x) = -f(x)\), the function has origin symmetry.

For our function \(6x^3 - 9x - x^5\):

• Substitute all x's with -x: \(f(-x) = -6x^3 + 9x + x^5\).
• Check if \(-f(x)\) or \(f(x)\) matches the computed expression.

Since \(f(-x) eq f(x)\) and \(f(-x) eq -f(x)\), the graph is neither symmetric about the y-axis nor the origin. Recognizing the absence of symmetry helps anticipate the overall layout of the graph.

End Behavior

The end behavior of a polynomial function describes the direction the graph heads as x approaches infinity or negative infinity. We observe end behavior primarily through the polynomial's leading term, which dictates how the function behaves.For our function with the leading term \(-x^5\),:

• Since the degree is 5 (odd), and the leading coefficient is -1 (negative), the graph will fall on both ends.
• This translates to \(f(x)\rightarrow -\infty\) as \(x\rightarrow \infty\) and \(f(x)\rightarrow \infty\) as \(x\rightarrow -\infty\).

Understanding the expected behavior at the ends of the graph helps when sketching the function, ensuring its shape makes sense with calculated intercepts and turn points.