Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using a Graphing Utility to Graph an Equation In Exercises \(31-44,\) use a graphing utility to graph the equation. Use a standard viewing window. Approximate any \(x\) - or \(y\) -intercepts of the graph. \(2 y-x^{2}+8=2 x\)

Step-by-Step Solution

Answer

After graphing with a graphing utility, we've got to the answer with x-intercepts and y-intercepts. The actual values will depend on the graph, be sure to record them for all calculations.

Step 1: Rearrange the equation

The first step is to rearrange the equation \(2y - x^2 + 8 = 2x\) to the standard form \(y = f(x)\). Move all other terms to the right-hand side, which gives us the equation: \(y = (x^2 + 2x - 8)/2\)

Step 2: Graph the equation using a graphing utility

Now that we have the equation in a standard form, utilize a graphing utility to draw the graph. Make sure to use a standard viewing window.

Step 3: Find x-intercepts

X-intercepts are the points where the graph touches or crosses the x-axis, meaning y = 0. In this graphing utility, set \(y = 0\) and solve the equation to find the x-intercepts.

Step 4: Find y-intercepts

Similar to x-intercepts, y-intercepts are points where the graph touches or crosses the y-axis, meaning \(x = 0\). Set \(x = 0\) and solve the equation in the graphing utility to find the y-intercepts

Key Concepts

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or software, that helps you visualize mathematical equations by plotting them on a coordinate grid. This visualization makes it easier to understand the behavior of equations and their components. When using a graphing utility, you input an equation, which it then converts into graphical form.
For the given equation \[2y - x^2 + 8 = 2x,\]rearranging it into \[y = \frac{x^2 + 2x - 8}{2}\]allows for direct input into the utility. Remember to use a standard viewing window, typically set to show a range of x and y values that adequately display the key features of the graph.
Graphing utilities often include features to identify intercepts, maxima, minima, and intersections, which are crucial to understanding the properties of the graphed equation. These features simplify your work by offering a more precise analysis than hand-drawn graphs.

X-intercepts

The x-intercepts of a graph are the points where the graph crosses or touches the x-axis. This means at these points, the y-value is equal to zero. To find the x-intercepts using a graphing utility:

• Input the rearranged equation: \[y = \frac{x^2 + 2x - 8}{2}\]
• Set the y-value to zero in the utility
• Solve the equation for x

This will provide the x-values where the graph meets the x-axis.
Finding the x-intercepts can often involve solving a quadratic equation, which might produce multiple intercepts or none, depending on the nature of the equation. In the visual output of the graphing utility, these intercepts can be spotted as the points where the line or curve meets the x-axis, serving as visual checks for your calculations.

Y-intercepts

Y-intercepts occur where the graph intersects or touches the y-axis. At this point, the x-value is zero. To determine the y-intercepts through a graphing utility:

• Input the rearranged equation: \[y = \frac{x^2 + 2x - 8}{2}\]
• Substitute x with zero in the utility
• Observe the resulting y-value

The y-intercept is the y-coordinate at the point where the graph crosses the y-axis.
Y-intercepts are often easier to find than x-intercepts due to the simplicity of setting x to zero and solving the equation for y. In the graph, this intercept is typically represented by a point on the y-axis, providing an easy reference for confirming the calculated value visually.