Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\)

Step-by-Step Solution

Answer

The function \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\) has a y-axis symmetry and the x-intercepts are at \(t = -4\) and \(t = 4\).

Step 1: Identify the type of function

The function \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\) is a polynomial of degree 4. Its degree is even, and the leading coefficient is negative.

Step 2: Graph the function

Plotting \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\) on a graph. From the plotted graph, symmetry can be observed.

Step 3: Identify symmetry

The graph of the equation is symmetric with respect to the y-axis. This is because when we replace \( t \) with \(-t\) in the equation, the equation remains the same.

Step 4: Identify the x-intercepts

The x-intercepts of a function are the values of x for which the function equals 0. Therefore, setting \(g(t)=-\frac{1}{2}(t-4)^{2}(t+4)^{2}\) equal to zero and solving for \( t \), the x-intercepts are -4 and 4.

Key Concepts

Symmetry in Polynomial Functions

When examining the symmetry of polynomial functions, it's important to understand how the function behaves when its variable is replaced with its negative counterpart. For the function \( g(t) = -\frac{1}{2}(t-4)^{2}(t+4)^{2} \), replacing \( t \) with \( -t \) results in an identical function: \( g(-t) = g(t) \).
This indicates that the function is symmetric with respect to the y-axis.

Y-axis symmetry implies that for every point \((t, g(t))\) on the graph, the point \((-t, g(t))\) also exists on the graph.

• To check for y-axis symmetry in general, replace \( t \) with \( -t \) and see if you obtain the same expression.
• If y-axis symmetry is present, it suggests that the graph of the function will be a mirror image on either side of the y-axis.

Understanding symmetry helps in predicting the behavior of the polynomial function without actually plotting every point.

Finding X-Intercepts of a Polynomial

X-intercepts are the points where the graph of the function crosses the x-axis. For any polynomial like \( g(t) = -\frac{1}{2}(t-4)^{2}(t+4)^{2} \), finding the x-intercepts involves setting the function equal to zero. Essentially, we are solving the equation \( -\frac{1}{2}(t-4)^{2}(t+4)^{2} = 0 \).

To solve this, we set each factor equal to zero:

• \((t-4) = 0\) gives \( t = 4 \)
• \((t+4) = 0\) gives \( t = -4 \)

Thus, the x-intercepts are \( t = 4 \) and \( t = -4 \).
Knowing the x-intercepts gives us crucial points that help in sketching the graph and in understanding the behavior of the polynomial at those points.
Remember, x-intercepts are where the output of the function is zero.

Using Graphing Utilities

Graphing utilities can be powerful tools to visualize the behaviors of polynomial functions. They provide a quick and efficient way to plot complex functions like \( g(t) = -\frac{1}{2}(t-4)^{2}(t+4)^{2} \). By inputting the function into a graphing calculator or software, one can observe the shape, symmetry, and intercepts of the graph.

Benefits of using graphing utilities include:

• Immediate visualization of the function’s behavior.
• The ability to easily spot features such as symmetry and intercepts.
• Zooming in and out to closely examine specific parts of the graph.

Using technology, students can confirm manual calculations and get a deeper understanding of polynomial functions. This extends beyond simple plotting, allowing examination of transformations and other characteristics with ease.

Understanding the Degree of a Polynomial

The degree of a polynomial is a key factor in determining its shape and behavior. For the given function \( g(t) = -\frac{1}{2}(t-4)^{2}(t+4)^{2} \), the degree is 4. This results from the factors \((t-4)^{2}\) and \((t+4)^{2}\), which each contribute a degree of 2.

An even-degree polynomial, such as this degree 4 polynomial, typically has a symmetric behavior with respect to the y-axis and a rise/drop pattern in the end behavior according to the leading coefficient. Since the leading coefficient is negative in \( g(t) \), this means:

• The graph will start high and end high, or vice versa, depending on the view.
• Typically, for even-degree polynomials, extreme ends trend in the same direction.

Grasping the concept of degree not only helps in graphing and problem-solving but also in predicting how the polynomial function behaves to infinity. Determining how the degree affects symmetry and intercepts adds insight into the comprehensive understanding of polynomial functions.