Problem 48 - 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. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

Step-by-Step Solution

Answer

End behavior: rise to left and rise to right. \(x\)-intercepts at \(x = \pm 1\) with the function crossing the \(x\)-axis at these points. \(y\)-intercept at \((0, \frac{1}{2})\). Graph has origin symmetry and 3 turning points.

Step 1: Use the Leading Coefficient Test

The leading coefficient test tells us the end behavior of the function. In this function \(f(x) = \frac{1}{2}-\frac{1}{2} x^{4}\), the leading term is -\(\frac{1}{2}x^{4}\). Since the power is an even number and the coefficient is negative, the graph will rise to the left and rise to the right.

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

The \(x\)-intercepts can be found by setting \(f(x)\) equal to 0 and then solving for \(x\). This gives \(0 = \frac{1}{2}-\frac{1}{2} x^{4}\). Solving this equation yields \(x = \pm \sqrt[4]{1}\), which simplifies to \(x = \pm 1\). The function crosses the \(x\)-axis at these points.

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

The \(y\)-intercept can be found by setting \(x\) equal to 0. This gives \(f(0) = \frac{1}{2}\), which means the function intercepts the \(y\)-axis at \((0, \frac{1}{2})\).

Step 4: Determine the graph’s symmetry

This function has \(y\)-axis symmetry if it is unchanged when \(x\) is replaced with \(-x\). It has origin symmetry if the sign is changed when \(x\) is replaced with \(-x\). For this function, when you replace \(x\) with \(-x\), it does change the sign but when you replace \(x\) with \(-x\), it does not remain unchanged. Hence, the function has origin symmetry.

Step 5: Plot the graph

By using the end behavior obtained in step 1, the intercepts obtained in steps 2 and 3, and the symmetry analysis from step 4, sketch the graph. There should be 4 - 1 = 3 turning points if the graph is drawn correctly.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a useful method for predicting the end behavior of polynomial functions. In essence, it tells us how the graph of the polynomial will behave as the input values become very large in both the positive and negative directions. Specifically, this test considers the leading term of the polynomial, which is the term with the highest power of the variable and its associated coefficient.

The leading term for the function f(x) = \(\frac{1}{2} - \frac{1}{2}x^4\) is -\(\frac{1}{2}x^4\). When the leading exponent is even, as it is in this case (4), and the leading coefficient is negative (-\(\frac{1}{2}\)), the test indicates that the graph will 'rise' to the left and 'rise' to the right. This means that both tails of the graph point upwards as x approaches positive and negative infinity.

x-intercepts of polynomials

The x-intercepts of a polynomial, also known as zeroes or roots, are the points where the graph of the polynomial intersects the x-axis. These points are found by setting the polynomial function equal to zero and solving for x. In the given function, solving \(0 = \frac{1}{2} - \frac{1}{2} x^4\) leads to the intercepts at \(x = \pm 1\).

Further, at each x-intercept, we can classify the intercept based on how the graph behaves: does it cross the x-axis, indicating a change in the sign of f(x), or does it merely touch the x-axis and turn around. For this function, since we have distinct roots (not repeated), the graph crosses the x-axis at these intercepts.

y-axis symmetry

A graph possesses y-axis symmetry if it looks the same on both sides of the y-axis. In other words, for every point on the graph, there is an identical point mirrored across the y-axis. To determine if a polynomial function has y-axis symmetry, we substitute -x for x and see if the resulting function is equivalent to the original. If the function remains unchanged, the graph is symmetric about the y-axis.

In our example, when we replace x with -x in \(f(x) = \frac{1}{2} - \frac{1}{2}x^4\), the function does not remain unchanged. This demonstrates the absence of y-axis symmetry for this function.

origin symmetry

Origin symmetry occurs in a graph when a function is mirrored across both the y-axis and the x-axis. For a function to have origin symmetry, replacing x with -x in the function should yield the original function negated. It implies that for any point (x, y) on the graph, there will be a corresponding point (-x, -y).

The function \(f(x) = \frac{1}{2} - \frac{1}{2}x^4\) does exhibit origin symmetry since substituting -x for x results in a sign change for the entire function, effectively mirroring points across both axes. Consequently, origin symmetry is confirmed for this polynomial function.