Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x+\frac{8}{x-5}-2$$

Step-by-Step Solution

Answer

The function is graphed, the x-intercepts are approximated from the graph, then these intercepts are calculated exactly by setting \(y=0\) in the original function. The results of both methods are compared, and they should match for confirmation.

Step 1: Graph the function

Firstly, plug the equation \(y=2 x+\frac{8}{x-5}-2\) into a graphing utility. Note down the shape of the graph and where it intersects with the x-axis.

Step 2: Approximate the x-intercept from the graph

Next, observe the points where the graph crosses the x-axis. These are the x-intercepts. Approximate their values.

Step 3: Solve the function with \(y=0\)

Now, set \(y=0\) in the given equation and solve for \(x\). This gives the exact x-intercepts of the function mathematically. So, plug \(0\) for \(y\) and solve the resulting equation \(0=2 x+\frac{8}{x-5}-2\).

Step 4: Compare the results

Finally, compare the x-intercept results from step 2 (approximated from the graph) and step 3 (calculated mathematically). Both should be the same if the graph is accurate and the mathematical solution is correct.

Key Concepts

Understanding x-intercepts

The term "x-intercepts" refers to the points where a graph crosses the x-axis. At these points, the value of the function is zero, meaning that the y-value is zero. Identifying x-intercepts is crucial when analyzing the behavior of a graph.

• To find the x-intercepts, you need to set the equation of the function equal to zero.
• Solve this equation to find the x-values that make the function zero.

For example, in the exercise given with the function \(y=2x+\frac{8}{x-5}-2\), setting \(y=0\) results in solving \(0=2x+\frac{8}{x-5}-2\), which gives the x-intercepts of the function. Understanding where these points are helps to understand the general solution and behavior of the function.

Using a graphing utility

A graphing utility is a tool that enables you to visualize mathematical functions by plotting them on a graph. This allows you to see the shape and key features of functions, such as their intercepts, asymptotes, and points of intersection with axes.

• Enter the equation into the graphing utility to view the graph.
• Use zoom tools to inspect different parts of the graph closely.

Graphing utilities can be software applications, calculators, or online tools. In our problem, the graphing utility allows us to plot \(y=2x+\frac{8}{x-5}-2\) to discover its shape and approximate the x-intercepts by observing where the graph crosses the x-axis. This visual approach provides a helpful way to check our mathematical solutions.

Steps to solve equations

Solving equations involves finding the value of the variable that makes the equation true. This can be done by following systematic steps to isolate the variable of interest. Let's outline steps typically followed:

• Start by simplifying the equation, if possible, by combining like terms or using algebraic techniques.
• When dealing with fractions or complex expressions, find a common denominator to simplify.
• Rearrange the equation to isolate the variable.
• Solve for the variable by performing inverse operations.

In our example, setting \(y=0\) in the function \(y=2x+\frac{8}{x-5}-2\), we solve for \(x\) to determine the precise x-intercepts. This process involves finding common denominators and solving linear equations, reinforcing your understanding of algebraic manipulation needed to find solutions.

Interaction between functions and graphs

Functions and graphs interact by translating algebraic expressions into visual representations. Understanding this relationship helps students to contextualize mathematical equations in a spatial format. Each function has a unique graph shape that depends on its equation.

• Linear functions, like \(y=mx+c\), appear as straight lines.
• Rational functions, which include polynomials divided by other polynomials, may have curves, breaks, or asymptotes.
• Recognizing these visual patterns aids in predicting behaviors such as growth, decline, or periodicity.

The exercise with \(y=2x+\frac{8}{x-5}-2\) demonstrates how a rational function behaves on a graph. The graph indicates trends, intercepts, and any asymptotic behavior that isn't evident from the equation alone. Recognizing how functions transform into graph shapes fosters a deeper understanding of mathematical concepts and solutions.