Problem 44 - 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. The maximum number of turning points of the graph is 3 , see graph 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. \(f(x)=x^{4}-x^{2}\)

Step-by-Step Solution

Answer

The end behavior of the function \(f(x)=x^{4}-x^{2}\) is that as \(x\) approaches both positive and negative infinity, \(f(x)\) approaches positive infinity. The \(x\)-intercepts are \(0\), \(-1\), and \(1\), at which the graph crosses the \(x\)-axis. The \(y\)-intercept is \(0\). The graph has y-axis symmetry. The maximum number of turning points is 3, which agrees with the graph.

Step 1: Use Lead Coefficient Test to Determine End Behavior

The leading term of \(f(x)=x^{4}-x^{2}\) is \(x^{4}\). Since the degree (4) is even and the leading coefficient (1) is positive, we conclude that as \(x\) approaches both positive and negative infinity, \(f(x)\) approaches positive infinity.

Step 2: Find \(x\)-intercepts

To find the \(x\)-intercepts, set \(f(x)=0\). Solving the equation \(x^{4}-x^{2}=0\) gives \(x^{2}(x^{2}-1)=0\), which further gives \(x=0\), \(x=-1\), and \(x=1\). Thus, the graph crosses the \(x\)-axis at these three points.

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

To find the \(y\)-intercept, set \(x=0\) in \(f(x)\). Thus, \(f(0)= (0)^{4}-(0)^{2}=0\), so the \(y\)-intercept is \(0\).

Step 4: Check for Symmetry

The graph of \(f(x)=x^{4}-x^{2}\) is symmetric with respect to the \(y\)-axis because replacing \(x\) with \(-x\) yields the original function, \(f(-x)=(-x)^{4}-(-x)^{2} = x^{4}-x^{2}=f(x)\). Therefore, the graph has y-axis symmetry and not origin symmetry.

Step 5: Discuss Turning Points and Graph the Function

Given that the maximum number of turning points of the graph is 3, and from our calculations above we've found 3 intercepts (which also correspond to changes in direction of the curve), no need for additional points. Plot the three points identified \((0,0)\), \((-1,0)\), \((1,0)\) and consider the end behavior to draw the accurate graph of the function.

Key Concepts

Leading Coefficient Test

The leading coefficient test is a useful tool for predicting the end behavior of polynomial functions. The end behavior refers to what happens to the values of the function, or the output, as the input values become very large in either the positive or negative direction.

In the case of the polynomial function \(f(x) = x^{4} - x^{2}\), the leading term is \(x^{4}\). Knowing that the degree is 4, which is even, and the leading coefficient is 1, a positive number, allows us to conclude that as \(x\) approaches both positive and negative infinity, \(f(x)\) approaches positive infinity.

This means the function graph will rise on both the left and right ends, forming a U-shape.

X-Intercepts

Finding the \(x\)-intercepts of a polynomial function involves setting the function equal to zero and solving for \(x\). This tells us where the graph crosses the \(x\)-axis. For \(f(x) = x^{4} - x^{2}\), setting the equation to 0 and factoring gives us \(x^{2}(x^{2} - 1) = 0\).

Solving this equation results in \(x = 0, -1, 1\). Therefore, the \(x\)-intercepts occur at these points. What's interesting here is that at each of these intercepts, the graph crosses the \(x\)-axis rather than just touching it, which implies a change in direction of the graph.

Y-Intercepts

The \(y\)-intercept of a function tells you where the graph crosses the \(y\)-axis. To find it, substitute \(x = 0\) into the function and solve for \(y\). When we do this with \(f(x) = x^{4} - x^{2}\), it simplifies to \(f(0) = 0\).

This tells us that the graph passes through the origin, or the point \((0, 0)\), which is both the \(x\)-intercept and the \(y\)-intercept for this function.

Symmetry of Functions

A graph has symmetry if it mirrors around a certain line or point. To check if a function's graph is symmetric with respect to the \(y\)-axis, we replace \(x\) with \(-x\) in the function and check if the resulting expression is equal to the original function.

For \(f(x) = x^{4} - x^{2}\), substituting yields \(f(-x) = (-x)^{4} - (-x)^{2} = x^{4} - x^{2} = f(x)\). Thus, this function is symmetric about the \(y\)-axis.

This symmetry implies that the graph looks the same on both sides of the \(y\)-axis.

Turning Points of Graphs

Turning points of a graph are those points where the graph changes direction, which corresponds to local maxima or minima. The maximum number of turning points of a polynomial function is always one less than its degree.

In the polynomial function \(f(x) = x^{4} - x^{2}\), the degree is 4. Hence, the maximum number of turning points is 3. These turning points often occur near the \(x\)-intercepts or between them.

For this function, the graph turns around at \(x = 0, -1, 1\), reaffirming our \(x\)-intercepts calculations, and connects beautifully with end behavior predictions.