Problem 69 - Introductory and Intermediate Algebra for College Students 4th

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. $$y=-\frac{1}{2} x-5$$

Step-by-Step Solution

Answer

The short answer will depend on the two points selected from the graph, but it should confirm the slope of the line to be \(-\frac{1}{2}\), as given in the equation.

Step 1: Graph the Line

The equation for the line is \(y=-\frac{1}{2} x-5\). Key in this equation into the graphing utility and create the graph.

Step 2: Trace and Select Points

Using the TRACE feature, move along the line to identify the coordinates of two different points. Each point should have both an X and Y coordinate.

Step 3: Compute the Slope

Once you have your two points, use them to compute the line's slope using the slope formula \( m = \frac{y_2-y_1}{x_2-x_1} \). Where, \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are the coordinates of the second point. Subsitute these values into the formula and calculate the slope.

Key Concepts

Slope Calculation

When dealing with linear equations, understanding how to calculate the slope is crucial. The slope of a line provides important information about its steepness and direction. The formula to calculate the slope (\( m \) ) of a line that passes through two points,\((x_1, y_1)\) and \((x_2, y_2)\), is:

\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]

• The slope is positive if the line rises from left to right.
• The slope is negative if the line falls from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope is characteristic of a vertical line.

Careful selection of points along the line is essential to accurate slope calculation. By using tools like a graphing utility, it is easier to ensure precise coordinates are chosen.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves representing geometric figures in a coordinate plane. For linear equations like \(y = -\frac{1}{2}x - 5\), it helps in understanding the spatial relationship between various points and lines.

In this form, the equation is already in slope-intercept form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept (-5 in this equation). This means the line crosses the y-axis at -5.

• Knowing the y-intercept helps in quickly drafting a rough sketch of the line.
• Determining the slope-first method allows sampling additional points on the line easily.
• Coordinate geometry aids in calculating distances, midpoints, and verifying collinear points.

Coordinate geometry is not just about plotting points; it's a comprehensive framework to analyze and interpret the stories lines tell on a plane.

Graphing Utilities

Graphing utilities are powerful tools that assist in visualizing mathematical equations, particularly useful for linear equations. With a simple input of the equation, they can plot a precise graph, allowing for quick analysis.

Common features of graphing utilities include:

• Equation Input: Type in the equation to generate the graph.
• Trace Functionality: Move along a graphed line to find precise coordinates.
• Zoom: Examine different sections of the graph in detail.
• Multiple Graphing: View several equations on the same grid for comparison.

Using the TRACE feature, crucial points along the line can be easily identified, which is particularly useful when manually calculating the slope or verifying intersects. Graphing utilities streamline the learning and understanding of geometry by visual reinforcement and simplify complex computations.