Problem 103 - Precalculus Essentials

Question

Use a graphing utility to graph each equation. Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=\frac{3}{4} x-2$$

Step-by-Step Solution

Answer

The computed slope of the line \(y = \frac{3}{4}x - 2\) using points (0,-2) and (4,1) is \(\frac{3}{4}\), which matches with the coefficient of \(x\) in the equation.

Step 1: Plot the graph

Plot the graph of the equation \(y = \frac{3}{4}x - 2\) using a graphing utility. Choose a scale that allows you to identify points on the line easily.

Step 2: Identify two points

Using the TRACE feature, move along the line on the graph and identify two points. Note down their coordinates. Suppose, for illustration, the points are (0,-2) and (4,1)

Step 3: Compute the slope

The formula for slope is \(\frac{y_2 - y_1}{x_2 - x_1}\). Using the coordinates of the two points chosen (0,-2) and (4,1), compute as follows: \(\frac{1 - {-2}}{4 - 0} = \frac{3}{4}\)

Step 4: Check the slope

The computed slope should be equal to the coefficient of \(x\) in the equation, which is \(\frac{3}{4}\). Since the computed slope and the slope from the equation are equal, the result is verified correct.

Key Concepts

Slope of a Line

The slope of a line indicates how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. This is expressed with the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
In the context of linear equations in the format \( y = mx + b \), \( m \) represents the slope. So, understanding and calculating the slope allows us to know and verify how the line is oriented in its path across a graph.

Trace Feature

The trace feature is a tool often found in graphing utilities that allows users to move along a graphed line to observe and record its coordinates.
It is particularly useful for identifying specific points on the line without manually calculating them. By using the trace feature, you can click and "trace" the graph to see the corresponding \(x\) and \(y\) values at different locations.
This direct technique is ideal for verifying any calculations such as confirming the coordinates of points used in slope calculations.

Using Graphing Utilities

Graphing utilities are tools like calculators or software used to visually represent mathematical equations and problems. They are incredibly helpful in solving, analyzing, and understanding the properties of mathematical entities like lines.
To use a graphing utility for plotting the line \( y = \frac{3}{4}x - 2 \), input the equation into the utility and visualize it. This visual representation can aid in better understanding the line's behavior and interaction with the grid, helping in confirming calculated slopes and identifying points quickly and efficiently.

Finding Coordinates

Finding coordinates on a graph involves identifying the \((x, y)\) positions of points. Using a graphing utility, once the line equation is graphed, you can find two points necessary for calculating the slope.
Begin by employing the trace tool or inspecting the plotted line to observe where the line intersects grid intersections, making the coordinates easier to define.
Document these points, such as something like \((0,-2)\) and \((4,1)\), and use them in further mathematical operations such as calculating the line's slope or checking other properties of the graph.