Problem 100 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=\frac{x}{x^{2}-5}$$

Step-by-Step Solution

Answer

The three points that lie on the graph of the equation are (-1, -0.25), (1, -0.25), and (2, -2).

Step 1: Choose Values for x

First, choose three different values for \(x\). Since there isn't any restriction in the problem, any real values except for those for which the denominator equals zero can be used. For instance, let's pick -1, 1, and 2.

Step 2: Substitute x=-1

Substitute \(x=-1\) into the equation: \(y=\frac{-1}{(-1)^{2}-5} = -\frac{1}{4}\). So, the first point is (-1, -0.25).

Step 3: Substitute x=1

Substitute \(x=1\) into the equation: \(y=\frac{1}{1^{2}-5} = -\frac{1}{4}\). Therefore, the second point is (1, -0.25).

Step 4: Substitute x=2

Finally, substitute \(x=2\) into the equation: \(y=\frac{2}{2^{2}-5} = -\frac{2}{1}\). The third point is (2, -2).

Key Concepts

Substitution Method

The substitution method is a crucial technique in algebra that allows you to find the value of one variable in terms of others by directly substituting numbers or expressions into an equation. In this method, you replace a variable with a given value or another expression that is equal to it. To employ the substitution method effectively, follow these steps:

• First, identify which variable needs to be substituted. In our exercise, we substitute various values for \(x\).
• Substitute these values into the equation to find the corresponding \(y\) values. Here, we substituted \(x = -1\), \(x = 1\), and \(x = 2\) into the equation \(y = \frac{x}{x^2 - 5}\).
• Ensure your substitutions avoid undefined expressions, like when the denominator in a fraction is zero.

This method helps convert equations into understandable numeric outcomes. It is especially useful for simplifying complex equations and making them manageable.

Graphing Equations

Graphing equations allows you to visually interpret algebraic relationships and understand the behavior of functions. It’s a powerful tool that helps see how changes in \(x\) affect \(y\). This visual approach can provide insights that are not easily detectable through purely algebraic manipulation.To graph an equation like \(y = \frac{x}{x^2 - 5}\):

• Choose a set of values for \(x\) that do not make the denominator zero, to avoid undefined points.
• Calculate the corresponding \(y\) values using the substitution method. We found points like (-1, -0.25), (1, -0.25), and (2, -2) by substituting specific values of \(x\).
• Plot these points on a coordinate plane, with \(x\) on the horizontal axis and \(y\) on the vertical axis.
• Connect the points to reveal the general shape of the graph.

Analyzing the graph, you'll notice curves, slopes, and possibly asymptotic behavior, all telling you more about the function's properties.

Analyzing Functions

Analyzing functions involves understanding the characteristics of the function based on its algebraic expression and graphical representation. This includes identifying key features like intercepts, domain, range, and asymptotes. For the function \(y = \frac{x}{x^2 - 5}\), consider the following:

• Intercepts: Find where the function crosses the axes by setting \(x\) or \(y\) to zero. In this case, for \(y = 0\), \(x\) must also be 0, giving an intercept at (0,0).
• Domain: Determine the values of \(x\) for which the function is defined. Here, the denominator \(x^2 - 5\) should not be zero, implying \(x eq \pm\sqrt{5}\).
• Asymptotes: As \(x\) approaches values that make the denominator zero, the function's value will increase or decrease without bound, showing vertical asymptotes at \(x = \pm\sqrt{5}\).
• Range: Consider all possible \(y\) values the function can take, which can be determined from the graph.

By thoroughly analyzing a function, you can predict its behavior in various situations and understand its comprehensive nature. This insight is crucial for problem-solving and applying algebraic concepts in real-world scenarios.