Problem 68 - 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. $$y=2 x+4$$

Step-by-Step Solution

Answer

The slope of the line given by the equation \(y=2x+4\) is 2.

Step 1: Graph the Equation

To begin with, input the equation \(y=2x+4\) into the graphing utility. This will provide a visual representation of the line corresponding to this equation.

Step 2: Find Two Points

Next, use the TRACE feature of the graphing utility which moves along the line and displays the coordinates of the points on the line. Identify two different points and note down their coordinates. Let's say we have points A and B with coordinates \((x1, y1)\) and \((x2, y2)\) respectively.

Step 3: Compute the Slope

Now, using the formula for calculating slope between two points, which is \(slope = (y2 - y1) / (x2 - x1)\), substitute the coordinates of the identified points. For instance, if \(x1=1, y1=6\) and \(x2=2, y2=8\), then the slope would be \( (8-6) / (2-1) = 2\), which should match the slope in the original equation.

Key Concepts

Slope Calculation

The concept of slope is fundamental when dealing with linear equations. The slope tells us how steep a line is and the direction it goes. When calculating the slope, we use the formula:

• The slope \( m \) is calculated as \( \frac{y_2 - y_1}{x_2 - x_1} \).

This formula essentially measures the ratio of the vertical change (``rise") to the horizontal change (``run") between two distinct points on the line.

By identifying two points on the line, we can calculate the slope and understand more about how the line behaves. A positive slope indicates the line rises as it moves from left to right, whereas a negative slope indicates it falls. In the exercise, the computed slope (2) means that for every unit increase in the x-direction, the y-value increases by 2 units.

Graphing Utility

Graphing utilities are powerful tools that allow you to visualize equations instantly. Imagine typing the equation \( y = 2x + 4 \) into the utility. What you get is an immediate graph of the line defined by the equation. These tools can be software programs on computers or apps on calculators, and are very helpful for students trying to understand the behavior of equations visually.

The graphing utility not only helps to plot the graph but often comes with a handy feature called TRACE. This feature allows users to navigate along the graphed line, giving the coordinates of any point on the line, which is essential when you're asked to find specific points or check your work.

Coordinates Identification

Finding coordinates on a line helps you to gain more insights about that line's specific points. In this exercise, after plotting the line using a graphing utility, it is essential to identify at least two coordinates to compute the slope. The TRACE feature makes this task straightforward as it shows the real-time x and y values as you move along the line.

Let's say the TRACE feature indicates the point (1, 6) when you navigate a little further, it then shows (2, 8). These coordinates are crucial because they are inputs in your slope calculation formula. For every linear equation,

• Choose two distinct points—not too close to ensure accuracy,
• Ensure these points lie on the line,

and be consistent in identifying the points as you work with different functions.