Problem 101 - College Algebra Essentials

Question

Use a graphing utility to graph each equation.Then use the TRACE feature to trace along the line and find the coordinates of two points Use these points to compute the line's slope. Check your result by using the coefficient of \(x\) in the line's equation. $$y=-3 x+6$$

Step-by-Step Solution

Answer

The slope of the line \(y = -3x + 6\) is -3, which is the same as the coefficient of \(x\) in the equation.

Step 1: Graph the Equation

Use a graphing utility to graph the line \(y = -3x + 6\). The graph should intersect the y-axis at \(y = 6\) and have a negative slope.

Step 2: Identify Two Points

Use the TRACE feature to identify two points on the line. Let's denote these points as \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\). The exact coordinates aren't crucial, as long as they are actual points on the line.

Step 3: Calculate the Slope

Compute the slope of the line using the formula for slope: \[m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\] Substitute the coordinates of \(P_1\) and \(P_2\) into formula to find the slope.

Step 4: Compare the Slope

Compare the calculated slope \(m\) with the coefficient of \(x\) in the line's equation, which is -3. They should be equal if all previous steps were done correctly.

Key Concepts

Slope Calculation

When graphing linear equations, one of the most critical concepts is the calculation of the slope, often denoted as 'm'. The slope indicates the steepness of the line and the direction in which the line moves. To calculate the slope when given two points, we use the following formula:
\[\begin{equation}m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\end{equation}\].
In a graphical representation, if we denote the two points as \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), we simply subtract the y-coordinate of the first point from the second, and do the same with the x-coordinates, placing our results in the numerator and denominator respectively. A positive result signifies an upward trend from left to right, while a negative result indicates a downward trend. If the slope is zero, it suggests a horizontal line, and an undefined (division by zero) result indicates a vertical line. Understanding how to calculate slope is essential for interpreting and graphing linear equations effectively.

Using Graphing Calculators

Graphing calculators are incredible tools for visualizing and understanding linear equations. They allow you to quickly plot the equation and analyze its features. To graph the linear equation \(y = -3x + 6\) using a graphing calculator, start by entering the equation into the calculator's graphing function. Once the line is plotted, you can use features like the 'TRACE' function to move along the line and identify the coordinates of specific points. This hands-on approach helps verify analytical solutions, such as slope calculations, by allowing you to see the linear relationship between variables and how it corresponds to the equation.
While using a graphing calculator, ensure you're familiar with how to adjust the viewing window to properly fit and display the graph of your equation. Mastery of these tools can simplify complex graphing tasks and support a better understanding of linear functions and other mathematical concepts.

Linear Equation Representation

A linear equation can be represented in various forms, but one of the most common is the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) the y-intercept. The given equation in the exercise \(y=-3x+6\) aligns with this structure, indicating a slope of -3 and a y-intercept of 6. This means that for every unit increase in \(x\), \(y\) decreases by three units, and when \(x=0\), the line crosses the y-axis at \(y=6\).
Understanding how to represent linear equations in this form is crucial for quickly identifying the characteristics of the line without needing to graph it every time. Moreover, the slope-intercept form paves the way for easy comparisons between different linear equations, as it highlights the rate of change and the initial value with direct clarity.