Problem 43 - College Algebra

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 uning points to check whether it is drawn correctly. $$ f(x)=x^{4}-9 x^{2} $$

Step-by-Step Solution

Answer

The end behavior of the graph is both ends pointing upwards. It crosses the \(x\)-axis at \(0, -3, 3\) and the y-intercept is \(0\). The graph doesn't have y-axis or origin symmetry.

Step 1: Apply the Leading Coefficient Test

The Leading Coefficient Test states that the end behavior of a polynomial function solely relies on the degree of the polynomial and the sign of its leading coefficient. In case of our function \( f(x)=x^{4}-9 x^{2} \), the leading coefficient is positive and the degree is even. Therefore, as \(x\) approaches positive or negative infinity, \(f(x)\) also approaches positive infinity.

Step 2: Determine the \(x\)-intercepts

The \(x\)-intercepts refer to the points where the graph crosses the \(x\)-axis. It can be found by setting \(f(x) = 0\). So solve \(x^{4}-9 x^{2} = 0\). Factor out an \(x^{2}\) to get \(x^{2}(x^{2} - 9) = 0\). Solving these gives \(x = 0, -3, 3\). The function will cross the \(x\)-axis at these points because the multiplicity of each root is odd.

Step 3: Find the \(y\)-intercept

The \(y\)-intercept is the point where the graph crosses the \(y\)-axis, which occurs when \(x = 0\). For our function, applying \(x=0\) gives \(f(0) = 0\), so there is a \(y\)-intercept at \(0\).

Step 4: Check for symmetry

The function \(f(x)\) has y-axis symmetry if \(f(-x) = f(x)\) and has origin symmetry if \(f(-x) = -f(x)\). Upon substituting, for our function neither conditions hold true. Hence, the graph neither has y-axis symmetry nor origin symmetry.

Step 5: Plot additional points and graph

First plot the intercepts and sketch the end behavior. Given the function is a 4th degree polynomial, the maximum number of turning points is 3. Choosing few additional points and joining them in a smooth curve that alternates direction at each intercept, we get the graph. Make sure the graph is a smooth curve with no sharp corners or discontinuities.

Step 6: Check if it's drawn correctly

Having drawn the graph, the function should cross the x-axis at \(x=0, -3, 3\) and have an y-intercept at \(y=0\). The function should not have any symmetry, and its ends should both point upwards.

Key Concepts

Leading Coefficient Test

When graphing polynomial functions, the Leading Coefficient Test is a quick way to determine how the graph behaves at the 'ends' of the x-axis. The end behavior of the function is influenced by both the degree of the polynomial and the sign of the leading coefficient, which is the coefficient of the term with the highest power.

For instance, for the function \( f(x)=x^{4}-9x^{2} \), the leading coefficient is 1 (implied, as it's the coefficient of \( x^4 \) term), and the degree is 4, which is even. Since the leading coefficient is positive, as \( x \) approaches both positive and negative infinity, the function will rise indefinitely. Thus, the ends of the graph head towards positive infinity, resembling a 'U' shape.

x-intercepts

The x-intercepts of a polynomial function are points where the graph cuts through or touches the x-axis. These points correspond to the roots of the equation obtained by setting the function equal to zero. For the given function \( f(x)=x^{4}-9x^{2} \), by solving \( x^{4}-9x^{2} = 0 \), we find the x-intercepts to be at \( x = 0, -3, 3 \).

Furthermore, because these roots are distinct and have odd multiplicities, the graph will cross the x-axis at these points. It’s crucial to distinguish whether the graph crosses or merely touches the axis because it affects the shape and the number of turns the graph makes.

y-axis symmetry

Symmetry in graphs of polynomial functions makes for an elegant visual property and can simplify graphing. A function has y-axis symmetry if it's unchanged when \( x \) is replaced with \( -x \). Formally, \( f(-x) = f(x) \). Origin symmetry exists if \( f(-x) = -f(x) \), resulting in a graph that is rotationally symmetric around the origin.

Our function \( f(x) \), when subjected to these symmetry tests, shows that it does not possess y-axis or origin symmetry. This is important as it tells us that the graph is not mirrored across the y-axis or the origin, giving a unique shape for positive and negative values of \( x \).

Turning Points

Turning points on the graph of a polynomial function are where the graph changes direction from increasing to decreasing or vice versa. The number of potential turning points is always one less than the degree of the function. Hence, for a fourth-degree polynomial like \( f(x)=x^{4}-9x^{2} \), we can expect up to three turning points.

Turning points are critical for understanding the overall shape of the graph. They enable the function to show complex behavior, such as hills and valleys that can be visualized when drawing the curve. When graphing, making sure to place turning points correctly ensures that the graph will represent the function accurately.