Problem 38 - 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. $$f(x)=x^{4}-x^{2}$$

Step-by-Step Solution

Answer

The function \(f(x) = x^{4} - x^{2}\) increases without bound for \(x \rightarrow \pm \infty\). It has x-intercepts at \(x=0\), \(x=1\), and \(x=-1\) and a y-intercept at \(y=0\). The graph has y-axis symmetry and has three turning points.

Step 1: Identify the end behavior

The leading term of the polynomial is \(x^{4}\), which is an even-degree term with a positive coefficient, so the end behavior of the function is that it rises to the right and to the left. Therefore, the end behavior is \((-\infty, 0)\) and \((\infty, 0)\).

Step 2: Find the x-intercepts

For finding the x-intercepts, set \(f(x) = 0\) and solve for \(x\). So, \(x^{4}-x^{2} = 0\). Factoring out \(x^{2}\) we get \(x^{2}(x^{2}-1) = 0\). Solving this we get \(x=0\), \(x=1\), and \(x=-1\). At \(x=1\) and \(x=-1\), the graph crosses the x-axis. At \(x=0\), it touches the x-axis and turns around.

Step 3: Find the y-intercept

To find the y-intercept, set \(x = 0\) in the function and compute \(f(0)\), which is 0. Therefore, the y-intercept is \(y=0\).

Step 4: Determine symmetry of the graph

To determine whether the graph has y-axis symmetry, origin symmetry, or neither, makes use of the fact that: A graph has y-axis symmetry if \(f(-x) = f(x)\). A graph has origin symmetry if \(f(-x) = -f(x)\). Checking these conditions for \(f(x)=x^{4}-x^{2}\) shows that the function has y-axis symmetry.

Step 5: Plot the graph and check for correctness

Now, after steps 1-4, we have a rough sketch of the graph. Now we will plot the graph of \(f(x) = x^{4} - x^{2}\) including the points \(x=0\), \(x=1\), and \(x=-1\). Also, from step 1 we know that the maximum number of turning points of the graph is \(n - 1 = 3\). By checking the graph, we can see that this holds true.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps us determine the end behavior of a polynomial function by examining the leading term's degree and coefficient. In the given function, \(f(x) = x^4 - x^2\), the leading term is \(x^4\). This term has an even degree and a positive coefficient. When a polynomial has an even degree and the leading coefficient is positive, the graph rises to both sides with behavior asymptotically approaching infinity as \(x\) moves towards positive or negative infinity. Conversely, if the leading coefficient was negative, the graph would fall on both ends. By understanding this, you can predict how the graph behaves far from the origin, which is particularly useful for sketching rough graphs.

x-intercepts

To find the \(x\)-intercepts of the polynomial function, you set the function equal to zero. For \(f(x) = x^4 - x^2\), set this equation to zero: \(x^4 - x^2 = 0\). Factor out \(x^2\), which yields \(x^2(x^2 - 1) = 0\). Solving \(x^2 = 0\) gives \(x = 0\), and solving \(x^2 - 1 = 0\) provides \(x = 1\) and \(x = -1\). Hence, the \(x\)-intercepts are \(x = 0\), \(x = 1\), and \(x = -1\).
Further analysis reveals the nature at each intercept:

• At \(x = 0\), the graph touches the \(x\)-axis and turns around, since \(x^2 = 0\) appears in the factored form \((x^2)^2\), indicating a tangent point.
• At \(x = 1\) and \(x = -1\), the intercepts imply that the graph crosses the \(x\)-axis, demonstrating a change in graph direction at these points.

y-intercept

The \(y\)-intercept of a function \(f(x)\) is found by evaluating the function at \(x = 0\). For \(f(x) = x^4 - x^2\), substituting \(x = 0\) gives \(f(0) = 0^4 - 0^2 = 0\). Thus, the \(y\)-intercept is \(y = 0\). This means the graph intersects the \(y\)-axis at the origin. Understanding \(y\)-intercepts is crucial as it provides an anchor point for graphing and corroborates the polynomial's intercept and symmetrical properties.

symmetry

Determining the symmetry of a function’s graph can simplify graphing and understanding its shape. A graph has y-axis symmetry if \(f(-x) = f(x)\) for all \(x\), meaning the graph mirrors about the \(y\)-axis. Origin symmetry occurs when \(f(-x) = -f(x)\), indicating a 180-degree rotational symmetry about the origin.
To decide the symmetry of \(f(x) = x^4 - x^2\), substitute \(-x\) into the function: \(f(-x) = (-x)^4 - (-x)^2 = x^4 - x^2\), which equals \(f(x)\). This verifies that the polynomial is symmetrical about the \(y\)-axis. Recognizing symmetry quickly provides insights into the function's behavior and simplifies plotting.

graphing polynomial functions

Graphing polynomial functions entails combining insights from the leading coefficient, intercepts, symmetry, and turning points. For the function \(f(x) = x^4 - x^2\), first note the end behavior rising to both sides, deduced from the even leading term. The \(x\)-intercepts are at \(x = -1, 0, 1\), with distinct crossing and touching behaviors.
Additionally, the \(y\)-intercept at \(0\) provides a starting plot point, while y-axis symmetry ensures uniform behavior on either side of the \(y\)-axis. Revealing the graph's symmetry helps predict reciprocal points without explicit calculation.

• Ensure a maximum of \(n - 1 = 3\) turning points, aligning with the even-degree polynomial rule.
• By synthesizing these features, draw an accurate and comprehensive representation of the polynomial’s graph.

Mastering these steps allows for effectively graphing a polynomial, capturing its intricate shape and behavior.