Problem 21 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. $$ g(x)=\frac{2}{x^{2}} $$

Step-by-Step Solution

Answer

The graph of \( g(x) = \frac{2}{x^2} \) is an even function with a vertical asymptote at \( x = 0 \) and approaches zero as \( x \) increases in magnitude.

Step 1: Identify the Function

The given function is a rational function defined as \( g(x) = \frac{2}{x^2} \). It is important to note this function is undefined at \( x = 0 \) since division by zero is undefined.

Step 2: Choose Values for x

Select a set of \( x \) values to create a table of values. It is often helpful to choose both positive and negative values as well as points near where the function is undefined. For this example, choose \( x = -3, -2, -1, -0.5, 0.5, 1, 2, 3 \).

Step 3: Calculate g(x) for Each x

Compute the value of \( g(x) \) for each chosen \( x \). Fill in the table of values:\[\begin{array}{|c|c|}\hlinex & g(x) \\hline-3 & \frac{2}{9} \-2 & \frac{2}{4} = 0.5 \-1 & 2 \-0.5 & 8 \0.5 & 8 \1 & 2 \2 & 0.5 \3 & \frac{2}{9} \\hline\end{array}\]

Step 4: Plot the Points

Using the table of values, plot each point \((x, g(x))\) on a coordinate grid. Remember, the function is not defined at \( x = 0 \), so no value exists for this point.

Step 5: Draw the Graph

Connect the plotted points to form a smooth curve. Observe the trend that the graph is symmetric with respect to the y-axis (even function) and that as \( x \) approaches 0 from both sides, \( g(x) \) tends toward \(+\infty\). As \( x \) increases or decreases in magnitude, \( g(x) \) approaches zero.

Key Concepts

Graph Sketching

Graph sketching is the process of drawing a rough outline of a graph based on certain key characteristics. For rational functions like \( g(x) = \frac{2}{x^2} \), this involves understanding behavior such as asymptotes, intercepts, and overall shape. Remember that this function is not defined at \( x = 0 \), resulting in a vertical asymptote at that point. The graph will approach infinity as \( x \) nears zero from both sides.

• Vertical asymptotes occur where the denominator is zero, creating a break in the graph.
• Horizontal asymptotes can be found by looking at limits as \( x \rightarrow \pm \infty \), which for this function is the x-axis since \( g(x) \) approaches zero.

For sketching, plot points from a table of values to determine the shape. Note that since \( g(x) \) involves \( x^2 \), the graph is symmetrical about the y-axis.

Table of Values

Creating a table of values is crucial for understanding how a function behaves across different points. In the example of \( g(x) = \frac{2}{x^2} \), choosing both negative and positive \( x \) values highlights how the function changes symmetrically. To generate this table, select values of \( x \) such as -3, -2, -1, -0.5, 0.5, 1, 2, and 3. Calculating \( g(x) \) for each of these values provides a set of coordinates to plot:

• For \( x = -3 \), \( g(x) = \frac{2}{9} \).
• For \( x = -2 \), \( g(x) = 0.5 \).
• Continue calculating similarly for other chosen values.

Tracking these points on a graph helps visualize the function. The careful selection of points also shows trends as \( x \) gets very small or very large.

Symmetry in Functions

Symmetry is a vital concept in understanding certain types of functions. The function \( g(x) = \frac{2}{x^2} \) is an even function due to the presence of \( x^2 \) in the denominator. This means it is symmetric about the y-axis. To test for symmetry, substitute \( -x \) into the function and observe that \( g(-x) = g(x) \).

• This symmetry results in the left and right sides of the graph being mirror images.
• Even functions always have their graphs exhibiting this property about the vertical axis.

Understanding symmetry helps simplify graph sketching by confirming that calculated points from one side of the y-axis will reflect identically on the other.