Problem 112 - Introductory Algebra for College Students

Question

Use a graphing utility to graph in a standard viewing rectangle, \([-10,10,1]\) by \([-10,10,1]\). Then use the \([\text { TRACE }]\) feature to trace along the line and find the coordinates of two points. $$y=2 x-1$$

Step-by-Step Solution

Answer

Two points on the graph of the given function might vary based on which ones are selected during the trace. For instance, points like (-1, -3) and (1, 1) might be two possibilities. Note that any two points on the line would suffice for this exercise.

Step 1: Graphing the linear function

First, the expression in the exercise is a linear function represented in slope-intercept form, \(y=mx+b\), where \(m =2\) is the slope and \(b = -1\) is the y-intercept. This indicates that the line crosses the y-axis at -1 and the slope of the line is 2. Plug this function into the graphing utility and then graph it, setting the viewing rectangle to extend from -10 to 10 on both the x and y axes as per the exercise's instructions.

Step 2: Using TRACE to find coordinates

After graphing the equation, use the TRACE function. TRACE is a feature that allows you to move along the graph of an equation in a step-by-step fashion. You can usually initiate it by pressing a button labeled 'trace' on your graphing utility. Once you've started tracing, choose two points on the line to determine their coordinates. The coordinates will often be displayed at the bottom of the graphing screen when you pause the trace at the chosen points.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when it comes to graphing linear functions. It is expressed as \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept — the point where the line crosses the y-axis. For the given equation \(y = 2x - 1\), the slope is 2, and the y-intercept is -1. This means that for every unit you move right along the x-axis, the line will move up two units because the slope is positive. With the y-intercept at -1, we start our graph at the point (0, -1) on the y-axis.

Graphing the line starts with plotting the y-intercept and then using the slope to find another point. From (0, -1), we move one unit to the right (along the x-axis) and two units up (along the y-axis) to continue the line. This simple yet efficient formula allows for quick and accurate construction of linear graphs.

Graphing Utility Use

Graphing utilities, such as calculators or computer software, provide a visual representation of equations, making it easier to understand their properties. To graph the linear function using a utility, input the equation \(y = 2x - 1\) and set the viewing parameters. In this case, we're using a standard viewing rectangle defined as \([-10,10,1]\) by \([-10,10,1]\), effectively covering a range from -10 to 10 on both axes, with intervals of 1. This controlled environment ensures that our graph doesn't lose detail by being too zoomed in or out.

Graphing utilities often come with tools for analyzing the graph. One such tool might be a 'Zoom' feature that allows you to get a closer look at certain parts of the graph. This is especially useful for identifying specific values or for observing graph behavior near crucial points like intercepts or where the function might have a relative maximum or minimum.

Graph Tracing

The TRACE feature of a graphing utility is an interactive way to explore the details of a graphed function. After plotting a linear equation like \(y = 2x - 1\), activating TRACE allows you to move a cursor along the curve. As you follow the line, each position of the cursor corresponds to a particular set of x and y coordinates.

Using TRACE is particularly helpful for understanding how the function behaves between plotted points. For students, this is a practical tool for finding specific points on a graph, especially when the exact values do not fall on grid lines. Tracing can also aid in identifying patterns, such as the uniformity of the slope in a linear equation, and it also helps with cross-checking your work by verifying the points on the graph.

Linear Equation Graphing

Graphing a linear equation involves plotting its graph on a coordinate plane and drawing the line that represents the equation. For the equation given, \(y = 2x - 1\), two points are needed to draw the line. The slope-intercept form allows us to easily find the starting point, or the y-intercept, and from there we use the slope to determine a second point.

With these points found, draw a straight line through them, extending the line across the axes. This visually demonstrates the constant rate of change of y with respect to x, which is the defining characteristic of a linear equation. The graph shows the continuous set of solutions that satisfy the equation, allowing one to see how x and y values interact with one another in a straight-line relationship on the plane.