Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the equation. Use the trace feature of the graphing utility to approximate the unknown coordinate of each solution point accurate to two decimal places. (Hint: You may need to use the zoom feature of the graphing utility to obtain the required accuracy.) \(y=\left|x^{2}-6 x+5\right|\) (a) \((2, y)\) (b) \((x, 1.5)\)

Step-by-Step Solution

Answer

For point (a), after analysis, utility should provide the y coordinate for \(x = 2\). For point (b), it should provide the x coordinate for \(y = 1.5\). The exact values of these coordinates depend on the graphing utility used and can vary slightly due to rounding.

Step 1 - Plotting the Given Function

Enter the function \(y = |x^2 - 6x + 5|\) into the graphing utility and generate the function plot.

Step 2 - Finding the Y-coordinate for Point (a)

Use the trace function of the graphing utility to find the y-coordinate associated with \(x = 2\) to two decimal places. This will be achieved by entering 2 for value of x and then obtaining y.

Step 3 - Finding the X-coordinate for Point (b)

Next, set \(y = 1.5\) and use the trace function to find the x-coordinate associated with this. Where the line \(y = 1.5\) intersects the function, it's the wanted x-coordinate. This value should be found to the two decimal places.

Key Concepts

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions, equations, and data sets. These tools are typically software programs or calculators that allow for quick and accurate graph plotting. Entering equations like \(y = |x^2 - 6x + 5|\) lets you see the function's shape and behavior.

With a graphing utility, you can:

• Visualize complex mathematical concepts with ease.
• Instantly see the effects of changing parameters in an equation.
• Identify points where the function intersects the axes or other lines.

Often, these utilities have additional features like the ability to trace points or zoom in to gain further insights, which we'll explore in the next sections.

Trace Feature

The trace feature in a graphing utility lets you move along the curve of a graph and see the coordinates of specific points. This is particularly useful for finding unknown values in an equation, such as the y-coordinate when \(x = 2\) or the x-coordinate when \(y = 1.5\).

Here’s how the trace feature works:

• Enter the specific value you wish to trace, such as \(x = 2\) in this exercise.
• The utility calculates the corresponding y-value and displays it.
• This gives a precise readout of values accurate to decimal places necessary for your solution.

The trace feature allows exploration with accuracy, making sure your calculations are correct.

Zoom Feature

Sometimes, you may need to get a closer look at certain areas of your graph. This is where the zoom feature comes in handy, allowing you to focus on specific parts of the graph with more clarity.

Using the zoom feature involves:

• Selecting a section of the graph you want to examine more closely.
• Adjusting zoom levels to find optimal clarity and detail.
• Identifying critical points, such as intersections, that require more precision.

In this exercise, zooming helps achieve the accuracy needed, especially when determining where \(y = 1.5\) intersects with the function. A detailed view ensures you gather every important data point.

Absolute Value Function

An absolute value function like \(y = |x^2 - 6x + 5|\) involves taking the absolute value of the result of a quadratic expression. This affects the graph significantly, as it always reflects negative outputs to positive values.

Key characteristics include:

• The graph takes a V-shape or W-shape, typical of absolute value functions.
• It’s symmetrical concerning the y-axis, with a minimum point or points where the expression inside becomes zero.
• The absolute value affects where the graph intersects with certain y-values, making features like trace and zoom essential for precise graphing.

Understanding the absolute value function helps to predict the overall graph shape and behavior, aiding in more efficient problem-solving and analysis.