Problem 104 - 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. \(g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3}\)

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \frac{1}{8}(x+1)^{2}(x-3)^{3}\) has no symmetry with respect to the \(x\)-axis, \(y\)-axis, or the origin. There are two \(x\)-intercepts: \(x = -1\) and \(x = 3\).

Step 1: Graph the function

Start by plotting the function \(g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3}\) using a graphing utility. This will make the graph visible and facilitate the analysis of symmetry and identification of \(x\)-intercepts.

Step 2: Analyze Symmetry

Evaluating the function at \(x\) and at \(-x\) will indicate any symmetry. You have to compute \(g(-x)\) and compare it with \(g(x)\). If \(g(x) = g(-x)\) the function is symmetric with respect to the \(y\)-axis. If \(g(x) = -g(-x)\) it's symmetric with respect to the origin. If neither of these conditions are met, there's just no symmetry.

Step 3: Determine the \(x\)-intercepts

The \(x\)-intercepts are the values of \(x\) that make the function output zero (\(g(x)=0\)). From the equation \(g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3}\), the output will be zero if \(x+1 = 0\) yielding \(x= -1\) or \(x-3 = 0\) yielding \(x=3\). Hence, there are two \(x\)-intercepts.

Key Concepts

Function Symmetry

Graphing functions not only gives us a visual representation but also helps identify symmetry. Symmetry is essential because it provides insight into the behavior of the function without the need for extensive calculations. For function symmetry with respect to the y-axis, we evaluate the function at negative x-values. This means checking if for any given x, the equation holds true:

• If \( g(x) = g(-x) \), the function is symmetric with respect to the y-axis, indicating that flipping the graph over the y-axis results in the same graph.
• If \( g(x) = -g(-x) \), the function displays origin symmetry, meaning that a 180-degree rotation around the origin results in the same graph.
• If neither condition applies, then the graph does not have this kind of symmetry.

Checking these conditions for the function \( g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3} \) through substitution and comparison allows us to determine its symmetry or lack thereof.

X-Intercepts

X-intercepts provide us the points where the graph crosses the x-axis. These points are crucial as they indicate where the function's value changes sign from positive to negative or vice versa.
The x-intercepts are found by setting the function equal to zero and solving for x. This means finding the roots of the equation:

• From the function \( g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3} \), by setting it to zero \( g(x)=0 \), we can identify that the equation is satisfied when \( (x+1)^{2}=0 \) or \( (x-3)^{3}=0 \).
• Solving these equations gives the values of \( x = -1 \) and \( x = 3 \).
• Thus, there are two x-intercepts for this function, at \( x = -1 \) and \( x = 3 \).

These intercepts tell us where the graph touches or crosses the x-axis, crucial for understanding the function's behavior.

Graphing Utility

Utilizing a graphing utility is a powerful tool to visualize complex functions. It allows us to plot the function quickly to observe its behavior, including aspects like symmetry and intercepts.
With a graphing utility, complicated algebraic expressions can be translated into a visual context, making analysis more straightforward.

• Tools like graphing calculators or software help students and mathematicians see the overall shape and behavior of the graph without manually plotting numerous points.
• By using a graphing utility on \( g(x)=\frac{1}{8}(x+1)^{2}(x-3)^{3} \), we gain a comprehensive view of where the function has its intercepts, its symmetry properties, and how it behaves over different intervals.
• The graphing utility is especially handy for more complex functions where algebraic manipulation might be cumbersome.

This makes a graphing utility a valuable resource in understanding and analyzing functions efficiently.