Problem 51 - 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. \(f(x)=6 x^{3}-9 x-x^{5}\)

Step-by-Step Solution

Answer

The end behavior is that as \(x \rightarrow -\infty, y \rightarrow \infty\) and as \(x \rightarrow \infty, y \rightarrow -\infty\). The x-intercepts can be found by solving the equation \(x(6x^{2}-9-x^{4})=0\). The y-intercept is \(0\). The graph has neither y-axis symmetry nor origin symmetry. The graph of the function can be completed using these points and characteristics.

Step 1: Determine the End Behavior

For end behavior, look at the highest exponent's coefficient. This is \(6x^{3}\) and \(-x^{5}\) in our polynomial. The term with the highest exponent dictates the end behavior, that is the \(-x^{5}\) term. Hence, given that the power is odd and the coefficient is negative, as \(x\) approaches infinity (\(+\infty\)), \(y\) will approach negative infinity (\(-\infty\)) , and as \(x\) approaches negative infinity (\(-\infty\)), \(y\) will approach positive infinity (\(+\infty\)). In simpler terms, the left side of the graph will point up and the right side will point down.

Step 2: Finding the x-intercepts

To find when the function crosses the x-axis (\(x\)-intercepts), set \(f(x)\) equal to zero and solve for \(x\). Thus, \(6x^{3}-9x-x^{5}=0\). Factoring out \(x\) gives \(x(6x^2-9-x^4)=0\). Therefore the solutions are \(x=0\), and \(x=\) the roots of the equation \(6x^{2}-9-x^{4}=0\). We would have to use a numerical method or graphing tool to find these roots.

Step 3: Finding the y-intercept

To find when the function crosses the y-axis (\(y\)-intercept), set \(x=0\) in \(f(x)\). This gives \(f(0)=6*0^{3}-9*0-0^{5}=0\). Therefore, the y-intercept is \(0\).

Step 4: Determine the Symmetry

If the function \(f(x)\) equals to the function with all \(x\) replaced by \(-x\) (\(f(-x)\)), then it is symmetric to y-axis. Here, \(f(-x)=-6x^{3}+9x-x^{5}\) which is not \(f(x)\). Replace \(x\) with \(-x\) in \(f(x)\), if it equals to \(-f(x)\), it is origin symmetric. But, \(f(-x)\neq -f(x)\). So, the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Graph the Function

Then, plot the \(x\)- and \(y\)-intercepts on a graph. Draw the ends of the graph using the end behavior determined earlier. Use the maximum number of turning points (which is degree of the polynomial minus 1 - hence, 4 turning points) to check if it is drawn correctly. If necessary, find a few additional points to complete the graph.

Key Concepts

End Behavior

Understanding the end behavior of a polynomial function is crucial for predicting how the graph behaves as it moves towards infinity or negative infinity. The end behavior is primarily determined by the term with the highest degree in the polynomial, which in this case is \[-x^{5}\].Given that the coefficient is negative and the degree is odd:

• As \(x\) approaches positive infinity (\(+\infty\)), the graph pulls downward towards negative infinity (\(-\infty\)).
• Conversely, as \(x\) approaches negative infinity (\(-\infty\)), the graph rises up towards positive infinity (\(+\infty\)).

In simpler terms, visualize the left end of the graph pointing up and the right end pointing down. This characteristic ``long tail'' effect helps in sketching the graph and predicting its behavior.

X-Intercepts

Finding the \(x\)-intercepts of a polynomial function involves identifying the points where the graph crosses the \(x\)-axis. This occurs when \(f(x) = 0\). For the function \(f(x) = 6x^{3}-9x-x^{5}\), set:\[6x^{3} - 9x - x^{5} = 0\]Start by factoring out an \(x\):\[x(6x^2 - 9 - x^4) = 0\]From this, immediately we can see one intercept at \(x = 0\). For the roots of the equation \(6x^2 - 9 - x^4 = 0\), numerical methods or graphing tools may be necessary to identify these roots. Once calculated, check if the graph merely touches or crosses each intercept. The multiplicity of each root helps to determine this:

• If the multiplicity is odd, the graph crosses the \(x\)-axis.
• If even, it touches and turns around at the \(x\)-axis.

Graph Symmetry

Graph symmetry is about determining whether the graph reflects about the \(y\)-axis or the origin. To investigate symmetry in the polynomial function \(f(x)\):

• Y-axis Symmetry: If \(f(x) = f(-x)\), the graph is symmetric with respect to the \(y\)-axis. Here, \(f(-x) = -6x^3 + 9x - x^5\), which is not equal to \(f(x)\). Hence, not y-axis symmetric.
• Origin Symmetry: If \(f(-x) = -f(x)\), the graph has origin symmetry. Checking here, \(f(-x)\) does not equal \(-f(x)\). Thus, the graph does not have origin symmetry either.

This polynomial graph, therefore, lacks both \(y\)-axis and origin symmetry.

Y-Intercept

The \(y\)-intercept of a graph is where the function crosses the \(y\)-axis. To find the \(y\)-intercept, set \(x = 0\) in the function \(f(x)\). The given polynomial is:\[f(x) = 6x^{3}-9x-x^{5}\]When \(x = 0\):\[f(0) = 6(0)^3 - 9(0) - (0)^5 = 0\]Therefore, the \(y\)-intercept of this graph is at the origin \((0,0)\). This point is valuable for sketching and understanding the starting point of the graph from a vertical perspective.