Problem 69 - Introductory Algebra for College Students

Question

Use a graphing utility to graph each equation in Exercises \(67-70\). 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. $$y=-\frac{1}{2} x-5$$

Step-by-Step Solution

Answer

Slope calculation heavily depends on the points selected on the line. However, since our line is \(y=-\frac{1}{2} x-5\), the slope would be \(-\frac{1}{2}\), regardless of the points chosen.

Step 1: Graphing the Equation

To graph the equation \(y=-\frac{1}{2} x-5\), input the equation into the graphing utility. The output will be a visual representation of the line that the equation represents.

Step 2: Tracing Two Points

Next, use the TRACE feature of the graphing utility to select two points on the line. Let's say these points are \(P_1 (x_1, y_1)\) and \(P_2 (x_2, y_2)\). Note down the coordinates of these points.

Step 3: Calculating the Slope

Finally, use the formula for the slope of a line, which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the coordinates of points \(P_1\) and \(P_2\), calculate the slope.

Key Concepts

Equation Graphing

Graphing an equation involves creating a visual representation of it, usually in the form of a line or curve, on a coordinate plane. In this exercise, the equation is given as \( y = -\frac{1}{2}x - 5 \). This is a linear equation, which means it will graph a straight line. To plot this equation, you will typically input it into a graphing utility or calculator, which will then display the corresponding line on a grid. This is a useful way to see the entire solution set of the equation at a glance. For those working manually, remember to plot key points and consider the slope and intercept for accuracy.

TRACE Feature

The TRACE feature in a graphing utility is a simple yet powerful tool. It allows you to move along a graph of an equation and explore specific points. When you use the TRACE function, the graphing utility highlights points along the curve or line, and displays their coordinates on the screen. This helps in understanding the behavior of the graph at specific values of \( x \) and \( y \). In this exercise, tracing helps us select and identify specific coordinates

• Use the arrow keys to navigate and trace along the graph.
• Identify and record the coordinates of at least two points on the line.

This process provides the necessary data to perform calculations, such as finding the slope.

Slope Calculation

Finding the slope of a line is crucial in understanding its steepness and direction. The slope, represented by \( m \), can be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line. For the equation \( y = -\frac{1}{2}x - 5 \), once the TRACE feature has helped us identify two points, we substitute their coordinates into the formula. This calculation tells us the rate of change of \( y \) with respect to \( x \) as we move along the line. The negative value in this case confirms that the line descends as it moves from left to right due to a negative slope.

Coordinate Geometry

Coordinate geometry combines algebra and geometry to study positions, distances, and planes. In the context of this exercise, it helps us understand the graphical representation of linear equations in terms of their coordinates. By plotting the equation \( y = -\frac{1}{2}x - 5 \), we're working within the coordinate plane, where:

• The x-axis and y-axis intersect at the origin \((0,0)\).
• Coordinates are expressed as \((x, y)\) pairs.

This visual approach aids students in comprehending not only the solution sets of equations but also the relationships between various geometric entities such as intercepts, gradients, and areas. Coordinate geometry is fundamental for solving not just equations but also for proving theorems and analyzing geometric shapes.