Problem 37 - 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. $$f(x)=x^{4}-9 x^{2}$$

Step-by-Step Solution

Answer

End behavior: as \(x → ± ∞\), \(f(x) → ∞\). x-intercepts are \(0, -3, 3\) where the graph crosses the x axis. y-intercept is at \(0\). The graph has \(y\)-axis symmetry.

Step 1: Determine the End Behavior using the Leading Coefficient Test

The test indicates the direction the arms of the function will point as x approaches positive and negative infinity, depending on the leading coefficient and the degree of the leading term. Our leading coefficient here is \(1\) and degree is \(4\), which is even. When both the degree and the leading coefficient are positive, the graph rises to the right and rises to the left. So as \(x\) approaches infinity, \(f(x)\) approaches infinity, and as \(x\) approaches negative infinity, \(f(x)\) also approaches infinity.

Step 2: Find the x-intercepts

To find the \(x\)- intercepts, we set \(f(x) = 0\), which results in the equation \(x^{4}-9x^{2} = 0\). Factoring gives us \(x^{2}(x^{2}-9)=0\), which further simplifies to \(x^{2}(x-3)(x+3)=0\). Thus the x-intercepts are \(0, -3, 3\). For each root, the graph crosses the x-axis.

Step 3: Find the y-intercept

To find the \(y\)-intercept, we replace \(x\) with \(0\) in the function \(f(x)=x^{4}-9 x^{2}\). This gives us \(f(0)= (0)^4-9(0)^2 = 0\). Therefore, the \(y\)-intercept of the function is \(0\).

Step 4: Determine the Symmetry

A function has \(y\)-axis symmetry if \(f(-x) = f(x)\), origin symmetry if \(f(-x) = -f(x)\), and neither if it meets none of these conditions. Substituting \(-x\) for \(x\) in the equation produces \(f(-x)=(-x)^4-9(-x)^2 =(x)^4-9(x)^2 =f(x)\), meaning the function has \(y\)-axis symmetry.

Step 5: Find Additional Points and Verify the Graph

It's not necessary in this case since the graph can be accurately drawn with the given x-intercepts, y-intercept and the known behavior of a quartic polynomial function.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a crucial concept in understanding the end behavior of polynomial functions. It helps us predict how the graph behaves as \(x\) approaches positive or negative infinity. In the function \(f(x)=x^{4}-9x^{2}\), the leading term is \(x^4\), which has a leading coefficient of \(1\) and a degree of \(4\).
This implies two important things:

• Since the degree is even, both ends of the graph will move in the same direction.
• As the leading coefficient is positive, the graph will rise on both ends.

Therefore, as \(x\) approaches both positive and negative infinity, \(f(x)\) approaches infinity, indicating that both ends of the graph go upwards.

End Behavior

Understanding the end behavior of a polynomial function provides insight into how the graph behaves at extreme values of \(x\). In our polynomial \(f(x) = x^4 - 9x^2\):

• The degree of the polynomial is \(4\), which is even, meaning the ends of the graph will exhibit the same behavior.
• With a positive leading coefficient (\(1\)), the polynomial will rise to the right and also rise to the left.

This behavior can be visualized as a curve that opens upwards on both sides. Such end behavior is typical for even-degree polynomials with positive leading coefficients.

Intercepts

Intercepts are the points where the graph of a polynomial crosses the axes. They provide key reference points for sketching the function.

• **\(x\)-intercepts:** These occur where \(f(x) = 0\). Setting \(x^4 - 9x^2 = 0\) and solving, we factor to get \(x^2 (x-3)(x+3) = 0\). This results in \(x = 0\), \(x = 3\), and \(x = -3\). At each intercept, the graph crosses the \(x\)-axis.
• **\(y\)-intercept:** This occurs where \(x = 0\). Substituting \(x = 0\) into the polynomial gives \(f(0) = 0\), so the \(y\)-intercept is the origin, \((0,0)\).

Intercepts help define the critical points and the general shape of the graph, guiding the drawing process.

Symmetry

Analyzing the symmetry of a polynomial function provides additional insights into its graph. A function is symmetrical about the \(y\)-axis if \(f(-x) = f(x)\) for all \(x\). Checking symmetry:

• Substitute \(-x\) for \(x\) in the function: \(f(-x) = (-x)^4 - 9(-x)^2 = x^4 - 9x^2\).
• Since \(f(-x) = f(x)\), this shows the function has \(y\)-axis symmetry.

Functions with \(y\)-axis symmetry are mirrored across the \(y\)-axis, simplifying the graphing process by reflecting one side over the axis.

Graphing Polynomials

Graphing a polynomial like \(f(x) = x^4 - 9x^2\) involves combining all the information gathered from the previous steps. Consider these aspects:

• **End behavior:** Since both ends rise, we can anticipate the arms of the graph to go upwards.
• **Intercepts:** The \(x\)-intercepts are \(-3\), \(0\), and \(3\). The \(y\)-intercept is also at \(0\).
• **Symmetry:** The graph is symmetric about the \(y\)-axis, so it is mirrored across this axis.
• With these points and the knowledge of symmetry, draw the curve, ensuring it passes through the intercepts and respects the end behavior.

Keep in mind, the maximum number of turning points a polynomial of degree \(n\) can have is \(n-1\). For our function, the maximum is \(3\). This helps verify the accuracy of the drawn graph by confirming the expected shape.