Problem 45 - 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. $$f(x)=-x^{4}+16 x^{2}$$

Step-by-Step Solution

Answer

The end behavior of the function \(f(x)=-x^{4}+16 x^{2}\) is to fall both to the right and to the left. The graph crosses the x-axis at x = 0 and x = ±4 and the y-intercept is at 0. The graph has y-axis symmetry but not origin symmetry. The sketch of the function can be confirmed by plotting a few additional points.

Step 1: Determine the end behavior

The Leading Coefficient Test states that if the leading coefficient of a polynomial function is positive, and the degree is even, then the graph will rise to the right and rise to the left. If the degree is odd, then the graph will rise to the right and fall to the left. If the leading coefficient is negative, this behavior is reversed. In the function \(f(x)=-x^{4}+16 x^{2}\), the leading coefficient is negative, and the degree is 4 (which is even), so the end behavior of the graph is to fall to the right and fall to the left.

Step 2: Find the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for x:\n\n\(0=-x^{4}+16 x^{2}\)\n\nThis can be simplified and factored into:\n\n\(0=x^{2}(-x^{2}+16)\)\n\nSetting each factor equal to zero gives the x-intercepts at x = 0 and x = ±4. The graph crosses the x-axis at these intercepts because the multiplicity of each root is an odd number (1).

Step 3: Find the y-intercept

The y-intercept of the function is found by setting x = 0. Substituting x = 0 into the function gives:\n\n\(f(0)=-0^{4}+16*0^{2} = 0\)\n\nTherefore, the y-intercept is at 0.

Step 4: Determine symmetry

A function has y-axis symmetry if replacing \(x\) with \(-x\) gives the same function. A function has origin symmetry if replacing \(x\) with \(-x\) gives the opposite function. After replacing \(x\) with \(-x\) in \(f(x)=-x^{4}+16 x^{2}\), we obtain the same function, thus the graph has y-axis symmetry but not origin symmetry.

Step 5: Sketch the graph

The graph starts and ends in the same direction (downwards) due to its end behavior. It crosses the x-axis at x = 0 and x = ±4 and the y-axis at y = 0. The graph can therefore be sketched taking into account the symmetry about the y-axis and confirmed by plotting a few additional points if required.

Key Concepts

Leading Coefficient Test

When graphing polynomial functions, it's essential to understand how the graph behaves as the input values become very large or very small. This is known as the end behavior of the graph, and the Leading Coefficient Test provides a reliable method for determining it. The test uses two pieces of information: the leading coefficient and the degree of the polynomial. If the leading coefficient is positive and the degree is even, the graph will rise on both ends. Conversely, if the leading coefficient is negative, the graph will fall on both ends when the degree is even. When the degree is odd, the graph will have opposite behaviors at the ends, rising on one side and falling on the other, depending on the sign of the leading coefficient.

In our exercise, with the function \(f(x)=-x^{4}+16x^{2}\),the degree is 4 (an even number) and the leading coefficient is negative. According to the Leading Coefficient Test, this means the graph will fall as \(x\)approaches both positive and negative infinity, giving us a clear picture of the graph's end behavior.

x-intercepts of Polynomials

The points where a polynomial graph crosses or touches the x-axis are known as x-intercepts. Finding these points can provide significant insight into the graph's shape and behavior. To find the x-intercepts, we set the polynomial equal to zero and solve for \(x\).

For the function given in the exercise, \(f(x)=-x^{4}+16x^{2}\),we find the intercepts by factoring and setting the factors equal to zero. This results in intercepts at \(x=0\)and \(x=\pm4\).A crucial aspect of x-intercepts is their multiplicity, which affects how the graph interacts with the axis. An intercept with odd multiplicity means the graph crosses the axis, while an even multiplicity indicates that the graph touches the axis and turns around. In our example, each intercept has a multiplicity of 1, signaling that the graph crosses the x-axis at those points.

y-intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. This is found by setting \(x\)to zero in the polynomial function and solving for \(f(x)\).

For our polynomial function \(f(x)=-x^{4}+16x^{2}\),setting \(x\)to zero yields \(f(0)=0\),indicating that the y-intercept of this graph is at the origin (0,0). The y-intercept provides a starting point when drawing a graph and is crucial for understanding the function’s behavior at \(x=0\).

Symmetry in Graphs

Symmetry can greatly simplify understanding and graphing polynomials. There are two main types of symmetry to consider: symmetry with respect to the y-axis and symmetry with respect to the origin. A graph is y-axis symmetric if flipping it over the y-axis doesn't change the graph. This occurs when replacing \(x\)with \(-x\)in the equation yields the same result.

On the other hand, a graph has origin symmetry if rotating it 180 degrees about the origin yields the same graph, which happens when replacing \(x\)with \(-x\)results in the negation of the original function. In our function, \(f(x)=-x^{4}+16x^{2}\),replacing \(x\)with \(-x\)yields the same function, indicating that it has y-axis symmetry. Identifying symmetry helps in drawing the graph and understanding its behavior.

End Behavior of Polynomials

Understanding the end behavior of a polynomial function refers to knowing how the graph behaves as \(x\)grows very large in the positive or negative direction, essentially where the graph is heading as it extends to infinity. This behavior is primarily based on the degree of the polynomial and the sign of the leading coefficient. For our function \(f(x)=-x^{4}+16x^{2}\),the end behavior is determined by the Leading Coefficient Test. Since it’s a fourth-degree polynomial with a negative leading coefficient, the graph falls off to infinity in both directions, creating a 'downward' shape on both ends of the graph. This insight informs us about the limits of the graph on both ends, which is crucial for sketching it accurately.