Problem 44 - Algebra and Trigonometry

Question

In Exercises 41–64, 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 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 graph is that it rises to the right and left. The function crosses the x-axis at \(x = 0, 1, -1\). The y-intercept is at (0,0). The graph is symmetric about the y-axis and it has 3 turning points, which confirms the graph is correct.

Step 1: Determine the End Behavior

The end behavior of the graph is determined by the leading coefficient, in this case \(1\), and the degree of the polynomial, which is \(4\). Since the degree is even and the leading coefficient is positive, the graph of the polynomial rises to the right and to the left. That is, as \(x\) approaches infinity or negative infinity, \(f(x)\) also approaches infinity.

Step 2: Find the x-intercepts

Setting the function equal to zero and solving for \(x\), we get \(x^4 - x^2 = 0\). Factoring out an \(x^2\), we get \(x^2(x^2 - 1) = 0\). Setting each factor equal to zero gives \(x = 0, x = 1, x = -1\). All of these are real roots and the graph crosses the x-axis at each of these points.

Step 3: Find the y-intercept

The y-intercept is found by setting \(x\) equal to zero. Subbing \(x = 0\) into the function \(f(x) = x^4 - x^2\), we find \(f(0) = 0\). Therefore, the y-intercept is (0,0).

Step 4: Determine the symmetry

Start by replacing \(x\) with \(-x\) in the equation: \(f(-x) = (-x)^4 - (-x)^2 = x^4 - x^2\), which is equal to the original function, so the function is even, meaning it has y-axis symmetry. It's not odd (no origin symmetry) since \(f(-x)\) is not equal to \(-f(x)\).

Step 5: Graph the function

Plotting the intercepts and a few other points for good measure, then linking them together taking into account the end behaviors and the y-axis symmetry, we obtain the graph of the function. The graph has 3 turning points, which matches the fact that a 4th degree polynomial can have up to 3 turning points.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test helps us understand the end behavior of a polynomial function's graph, giving us a peek into what happens when the values of the variable get very large or very small. For our polynomial function \(f(x) = x^{4} - x^{2}\), we note that the leading term is \(x^{4}\) with a leading coefficient of \(1\). Since this coefficient is positive and the degree of the polynomial (4) is even, it indicates that the graph will "rise" both to the left and the right. Specifically, as \(x\) approaches positive or negative infinity, \(f(x)\) will approach positive infinity.

X-intercepts

X-intercepts are the points where the graph crosses or touches the x-axis. To find these intercepts for \(f(x) = x^{4} - x^{2}\), we set the equation equal to zero and solve for \(x\). This gives us:

• Setting \(f(x) = 0\) results in \(x^{4} - x^{2} = 0\).
• Factoring yields \(x^{2}(x^{2} - 1) = 0\).
• This simplifies to \(x = 0\), \(x = 1\), and \(x = -1\).

Each of these points is a real root, and the graph crosses the x-axis at these intercepts, signaling a change in the function's direction at each x-intercept.

Y-intercepts

Y-intercepts occur where the graph intersects the y-axis, essentially showing where the function crosses the line where \(x = 0\). For our polynomial \(f(x) = x^{4} - x^{2}\), we determine the y-intercept by substituting \(x = 0\) into the function. Doing so results in:

• \(f(0) = 0^{4} - 0^{2} = 0\).

Therefore, the y-intercept is (0,0). This means the function passes through the origin when it crosses the y-axis.

Symmetry

When inspecting a function's symmetry, we check two types: y-axis symmetry and origin symmetry. For our given function, we substitute \(-x\) into the function, yielding:

• \(f(-x) = (-x)^{4} - (-x)^{2} = x^{4} - x^{2}\)

Since \(f(-x) = f(x)\), the function is even, exhibiting y-axis symmetry. This symmetry implies that the graph is the same on the left side of the y-axis as it is on the right. The function is not odd—that is, it doesn't have origin symmetry—because \(f(-x) eq -f(x)\).

Graphing polynomial functions

Graphing a polynomial function like \(f(x) = x^{4} - x^{2}\) involves a few steps to accurately depict its shape:

• Plot the intercepts: Begin by plotting the x-intercepts at (0,0), (1,0), and (-1,0), and the y-intercept at (0,0).
• Check for symmetry and use it: Use the y-axis symmetry by ensuring that the left side of the graph reflects the right side.
• Consider the turning points: A 4th-degree polynomial can have up to 3 turning points which manifest as peaks or troughs, aiding in shaping the curve.
• Apply the Leading Coefficient Test: The graph should rise on both ends as it extends left and right.

By following these steps, you can sketch a smooth, accurate representation of the polynomial's behavior across the Cartesian plane.