Problem 68 - 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. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The calculated slope of the line graph of the equation \(y = -3x + 6\) is -3.

Step 1: Graph the Equation

First, we will input the equation \(y = -3x + 6\) into the graphing tool. A straight line graph will be generated, which represents all the solutions to the equation.

Step 2: Trace along the Line

With the line graph of the equation displayed, we will use the TRACE feature to trace along the line. While tracing, we will note two different points on the line. Let's say we select Point \(A(1, 3)\) and Point \(B(2, 0)\).

Step 3: Calculate the Slope

After obtaining two points on the line, we can calculate the slope. The slope formula is \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the values obtained from the graph into the formula: \(m = (0 - 3) / (2 - 1) = -3\).

Key Concepts

Slope Calculation

The slope of a line measures its steepness and direction. In mathematics, it is often represented by the letter \(m\). To find the slope, you need two points on the line.

The formula to calculate slope is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line. For example, if given points \(A(1, 3)\) and \(B(2, 0)\), substitute these into the formula:

• \( m = \frac{0 - 3}{2 - 1} \)
• \( m = \frac{-3}{1} \)
• \( m = -3 \)

This shows the line has a steep negative slope, indicating it goes downwards from left to right.

Graphing Tool

A graphing tool helps visualize equations, particularly lines and curves. Inputting the equation \( y = -3x + 6 \) into such a tool generates a straight line. This line represents all possible solutions for \(x\) and \(y\).

Using graphing tools can simplify understanding of equations by turning abstract concepts into visual representations.

• Easy input: Many graphing tools allow you to simply type in the equation.
• Visual aid: The tool instantly displays the line or curve of the given equation.
• Interactive: Some tools offer features like zooming or adjusting graph parameters.

These capabilities make graphing tools ideal for students to grasp how equations behave.

Coordinate Points

Coordinates are pairs of numbers that describe a specific location on a graph. Each point on a two-dimensional graph has an \( x \)-coordinate and a \( y \)-coordinate, written as \( (x, y) \).

Choosing points on a graph helps in determining characteristics like the slope.

• Accurate graphing: Select precise points to avoid miscalculations.
• Point naming: It is common to label them, like Point \( A \) or Point \( B \).
• Reflection of the equation: Ensure selected points lie on the graph line of the equation.

For the equation \( y = -3x + 6 \), points like \( A(1, 3) \) and \( B(2, 0) \) are examples of how point positions confirm their relation to the line.

Trace Feature

The trace feature in graphing utilities allows you to explore graphs interactively by moving along the line and observing values at different points. It is particularly useful for finding coordinates.

Here's how the trace feature can enhance understanding:

• Exploration: Easily navigate along the plotted line to observe various points.
• Instant information: As you trace, the coordinates are typically displayed on-screen.
• Verification: Confirm that chosen points lie accurately on the graph line.

Using trace, you can directly involve and see the relationships between the graphical representation and numerical solutions.