Problem 43 - Algebra and Trigonometry

Question

In Exercises 41–64, 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. $$f(x)=x^{4}-9 x^{2}$$

Step-by-Step Solution

Answer

End behavior: as \(x \rightarrow -\infty, f(x) \rightarrow +\infty\) and as \(x \rightarrow +\infty, f(x) \rightarrow +\infty\). X-intercepts: \(x = 0, x = 3\), \(x = -3\). Y-intercept: 0. The graph has y-axis symmetry and no origin symmetry. The function is correctly graphed with a maximum of 3 turning points.

Step 1: Determine the End Behaviour

The Leading Coefficient Test is used to find the end behavior of a graph based on the sign of the leading coefficient and the degree of the polynomial function. For the given function \(f(x) = x^4 - 9x^2\), the leading coefficient is 1 and the degree is 4. Since the degree is even and the leading coefficient is positive, the end behavior of the function is: as \(x \rightarrow -\infty, f(x) \rightarrow +\infty\) and as \(x \rightarrow +\infty, f(x) \rightarrow +\infty\).

Step 2: Find the x-intercepts

To find the x-intercepts, set the function \(f(x)\) equal to zero and solve for \(x\). \[x^4 - 9x^2 = 0\] The equation can be written as: \[x^2(x^2 - 9) = 0\] This gives the solutions \(x = 0, x = 3\), and \(x = -3\). So, the graph crosses the x-axis at \(x = 0, x = 3\), and \(x = -3\).

Step 3: Find the y-intercept

To find the y-intercept, substitute \(x = 0\) into the function. \[f(x) = (0)^4 - 9(0)^2 = 0\] So, the y-intercept is 0.

Step 4: Determine the Symmetry

To determine the symmetry, replace \(x\) with \(-x\). If \(f(-x) = f(x)\), then the graph has y-axis symmetry. If \(f(-x) = -f(x)\), then the graph has origin symmetry. \[f(-x) = (-x)^4 - 9(-x)^2 = x^4 - 9x^2\] So, the graph has y-axis symmetry because \(f(-x) = f(x)\). The graph does not have origin symmetry because \(f(-x) \neq -f(x)\).

Step 5: Graph the Function

To graph \(f(x) = x^4 - 9x^2\), plot the x-intercepts at \(-3, 0\), and \(3\), the y-intercept at \(0\), and the points derived from the symmetry \(y = -x^4 + 9x^2\). Connect these points with a smooth curve, remembering that the graph opens upwards because the leading coefficient is greater than zero. Finally, remember that the function will have at most \(4-1 = 3\) turning points, as indicated by the degree of the polynomial minus one. Check that the graph drawn has at most these many turning points.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is an essential tool for understanding the behavior of a polynomial function as it stretches towards infinity. In polynomial functions, the leading coefficient and the degree of the polynomial determine how the graph behaves at the far left and right ends. For the function \(f(x) = x^4 - 9x^2\), the leading coefficient is 1, and its degree is 4, indicating an even degree.

Since the leading coefficient is positive and the degree is even, the function behaves in such a way that both ends of the graph rise upwards. Therefore, as \(x \rightarrow -\infty\), \(f(x) \rightarrow +\infty\) and similarly, as \(x \rightarrow +\infty\), \(f(x) \rightarrow +\infty\). Understanding this test helps in predicting the broad shape of the polynomial curve without even graphing it.

x-intercepts

To find the x-intercepts of a polynomial, we need to solve the equation when the function equals zero. By doing this, we find where the graph crosses or touches the x-axis. For the function \(f(x) = x^4 - 9x^2\), we set \(f(x) = 0\). This equation simplifies to \(x^2(x^2 - 9) = 0\), offering solutions \(x = 0\), \(x = 3\), and \(x = -3\).

At each x-intercept, it is important to determine if the graph crosses the x-axis or just touches and turns around. In this case, the graph crosses the x-axis at each of the intercepts \(x = 0, x = 3,\) and \(x = -3\). This helps in sketching the accurate shape of the graph.

y-intercept

The y-intercept of a polynomial function is the point where the graph crosses the y-axis. It provides a reference point to start sketching the graph. To find the y-intercept, we substitute \(x = 0\) into the function. For \(f(x) = x^4 - 9x^2\), substituting \(x = 0\) returns:

\[f(0) = (0)^4 - 9(0)^2 = 0\]

This indicates that the y-intercept is \(0\). The graph crosses the y-axis at the origin, helping provide a central point for symmetrically plotting additional points.

Symmetry

Symmetry in graphs helps identify the repeating patterns, making the graphing process simpler. There are primary types of symmetries to check: y-axis symmetry and origin symmetry. To check for these, we substitute \(x\) with \(-x\) in the function. For our function \(f(x) = x^4 - 9x^2\), substituting gives:

\[f(-x) = (-x)^4 - 9(-x)^2 = x^4 - 9x^2\]

This results in \(f(-x) = f(x)\), showing that the function has y-axis symmetry, meaning the left side of the graph mirrors the right side. For origin symmetry, the condition would be \(f(-x) = -f(x)\), which is not the case here. Thus, the graph exhibits only y-axis symmetry.

Graph of a polynomial function

Graphing the function \(f(x) = x^4 - 9x^2\) involves integrating all the information gathered from the end behavior, intercepts, and symmetry. Start by plotting the crucial points:

- x-intercepts at \(x = -3, 0,\) and \(3\)
- y-intercept at \(y = 0\)

Due to y-axis symmetry, you will notice that the graph appears balanced around the y-axis. The end behavior indicates the graph approaches positive infinity at both ends, opening upwards like a traditional U shape.

We can anticipate up to \(3\) turning points, calculated as one less than the degree of the polynomial. Plotting these details and joining the dots smoothly gives you the desired polynomial graph. In conclusion, all these components collaborate to give an accurate visual of the polynomial's behaviors and features.