Problem 14 - Technical Mathematics with Calculus

Question

Find the slope and the \(y\) intercept for each equation, and make a graph. $$y=-3 x+2$$

Step-by-Step Solution

Answer

The slope of the equation is -3, and the y-intercept is 2. After plotting the y-intercept (0, 2) on the graph, use the slope to find another point (1, -1) and draw the line through these points.

Step 1: Identify Slope and Y-Intercept

The equation is already in slope-intercept form, which is \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept. For the equation \(y = -3x + 2\), the slope (m) is -3, and the y-intercept (b) is 2.

Step 2: Plot the Y-Intercept

Begin the graph by plotting the y-intercept (0, 2) on the y-axis. This is the point where the line crosses the y-axis.

Step 3: Use Slope to Find Another Point

Use the slope to find another point on the line. Slope is rise over run. Here, a slope of -3 means go down 3 units (rise) and to the right 1 unit (run). If starting at (0, 2), this would result in the point (1, -1).

Step 4: Draw the Line

Draw a line through the two points, which represents the graph of the equation \(y = -3x + 2\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations. This form is written as \(y = mx + b\), where \(m\) signifies the slope of the line, and \(b\) marks the y-intercept, which is where the line crosses the y-axis.

The beauty of this form lies in its straightforward nature. When we look at the equation \(y = -3x + 2\), it is immediately clear that the slope \(m = -3\) and the y-intercept \(b = 2\). This format simplifies the process of graphing because it gives you the starting point and the direction in which the line heads.

Y-Intercept

The y-intercept is the point where the line of a linear equation crosses the y-axis. It's found when \(x=0\).

For the equation \(y=-3x+2\), the y-intercept is \(2\). That's because when \(x\) is zero, \(y\) equals \(2\). To graph this, you locate the y-intercept on the y-axis first. It's an essential step as it serves as an anchor for your line. From there, using the slope, we can determine the line's direction and continue plotting.

Plotting Points

When you're plotting points to graph a linear equation, it's like connecting the dots. Start with the y-intercept, which we've identified as \(2\) for the equation \(y = -3x + 2\). Place a dot at \(0, 2\) on the graph.

Next, use the slope to find your next point. Remember, the slope tells you how much y changes for a one-unit change in x (rise over run). From the y-intercept, move according to the slope: in this case, down three units because the slope is \( -3\) and then right one unit. You will land on the point \(1, -1\), which you then plot. Connecting these two points gives you the complete graph of the equation.

Slope

Slope is a measure of how steep a line is, and it's expressed as the ratio of the rise (the change in y) over the run (the change in x). The slope dictates the angle and direction of the line on the graph.

For the given equation \(y = -3x + 2\), the slope is \( -3\). This negative slope indicates that the line falls as it moves from left to right. To apply the slope, you start from the y-intercept and move down 3 units (because the slope is negative) and 1 unit to the right. It's the slope that guarantees that even if we choose different points to start from, the line's steepness and direction remain consistent.