Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=2 x-\frac{8}{x}$$

Step-by-Step Solution

Answer

From the graph, the x-intercepts are found to be at \(x = 2\) and \(x = -2\). These are confirmed when the function is set to 0 and solved for x, giving the solutions \(x = 2, -2\).

Step 1: Graphing the Function

The first step is to graph the function \(y = 2x - \frac{8}{x}\) using a graphing utility. This will give a visual representation of the function.

Step 2: Identifying the X-Intercepts from the Graph

The x-intercepts of a graph are the points at which the graph crosses or touches the x-axis. Looking at the graph of our function, the x-intercepts can be identified.

Step 3: Confirming the X-Intercepts with Algebra

To confirm the x-intercepts found from the graph, set \(y = 0\) in the function, resulting in the equation \(0 = 2x - \frac{8}{x}\). Next, solve this equation for x.

Key Concepts

Understanding X-Intercepts

The concept of x-intercepts is essential for understanding the behavior of a function. These are the points where the graph of a function crosses the x-axis.
In simpler terms, they are the values of \(x\) that make the function's output \(y\) equal to zero. Identifying these points can help us understand where a function changes direction or approaches a different value.In the function \(y = 2x - \frac{8}{x}\), setting \(y = 0\) helps find the x-intercepts. This results in equation \(0 = 2x - \frac{8}{x}\), which must be solved algebraically to determine the specific x-values where the graph intersects the x-axis.
By solving this equation, we confirm the points where \(y = 0\), giving us the exact positions of our x-intercepts.

Using a Graphing Utility

A graphing utility is a powerful tool that allows us to visualize mathematical functions. These tools, which can be physical calculators or software applications, help simplify complex equations by providing a visual representation.
Using a graphing utility for the function \(y = 2x - \frac{8}{x}\), you can quickly generate its graph to observe how it behaves over different values of \(x\).Here's how you can use a graphing utility effectively:

• Enter the function into the utility by typing \(y = 2x - \frac{8}{x}\).
• Adjust the viewing window to get a clear and detailed visual of the function.
• Observe where the graph crosses the x-axis, indicating potential x-intercepts.

These steps allow you to easily identify the approximate values of x-intercepts without manually solving the equations.

Algebraic Confirmation of X-Intercepts

After visually identifying the x-intercepts using a graphing utility, it's important to confirm these findings through algebra. Algebraic confirmation removes any doubt about inaccuracies due to visual estimation.To confirm the x-intercepts for \(y = 2x - \frac{8}{x}\), setting \(y = 0\) gives us the equation \(0 = 2x - \frac{8}{x}\). This simplifies to finding common values for \(x\).
Solving the equation algebraically involves:

• Multiplying through by \(x\) to eliminate the fraction: \(0 = 2x^2 - 8\).
• Solving for \(x\): First, rearrange to get \(2x^2 = 8\), then divide by 2 to get \(x^2 = 4\).
• Taking the square root of both sides results in \(x = \pm 2\). Thus, the x-intercepts are \(x = 2\) and \(x = -2\).

This step confirms what you saw on the graph, providing both numerical and visual certifications.