Problem 66 - Algebra 2

Question

Use the given information to graph each line. slope \(=-\frac{3}{2},\) through \((1,-4)\)

Step-by-Step Solution

Answer

To graph the line, plot the point (1, -4) and from there go down 3 units and right 2 units to plot another point. Draw the line through these points.

Step 1: Plot the initial point

Begin by plotting the point (1, -4) on the coordinate plane. This point will be used as a starting point to apply the slope and draw the line.

Step 2: Apply the slope

Since the slope is \( -\frac{3}{2} \), from the initial point (1, -4), move down 3 units vertically (the rise), and then 2 units to the right horizontally (the run). Plot a second point at the new location.

Step 3: Draw the line

Using a straightedge, connect the initial point with the new point. Since a line extends infinitely in both directions, continue to draw the line beyond the points ensuring it is straight, and includes the two points plotted. This line represents the graph of the equation with a slope of \( -\frac{3}{2} \) passing through the point (1, -4).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when graphing linear equations. It is one of the most straightforward ways to express a linear equation. The slope-intercept form is written as \(y = mx + b\), where \(m\) stands for the slope of the line, and \(b\) represents the y-intercept—the point where the line crosses the y-axis.

In the context of the given exercise, once you know the slope \(m = -\frac{3}{2}\) and the point through which the line passes \(1, -4\), you can calculate the y-intercept \(b\) using the equation \(y = mx + b\). Substitute the x and y values from the point into the equation and solve for \(b\). Once you have \(b\), you have the full equation of the line that can be graphed. This standard form makes plotting lines manageable because you start by plotting the y-intercept \(b\) and then use the slope \(m\) to find other points on the line by moving vertically and horizontally from the y-intercept.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface with perpendicular axes; namely, the horizontal x-axis and the vertical y-axis. The intersection of these axes is the origin, labeled as \(0, 0\). The plane is divided into four quadrants, with each quadrant representing a pair of sign values for \(x\) and \(y\).

When graphing an equation on the coordinate plane, such as in our exercise, you start by plotting a known point. As you gain more experience, you'll understand how the location of the point relates to its coordinates. For example, the point \(1, -4\) lies in the fourth quadrant since its x-coordinate is positive and its y-coordinate is negative. Understanding the coordinate plane helps with determining the direction in which to apply the slope, as was done in the exercise by moving down and to the right from the initial point.

Slope of a Line

The slope of a line represents the steepness or the incline of the line and is usually denoted by \(m\). It's a measure of how much the y-value of the line increases or decreases as the x-value increases. Calculated as \(m = \frac{rise}{run}\), the slope can tell us if the line is increasing, decreasing, horizontal, or vertical.

In a graphical sense, 'rise' refers to the movement up or down on the y-axis, while 'run' refers to the movement left or right on the x-axis. A positive slope means the line rises as it goes from left to right, and a negative slope means it falls. In our example, the slope is \( -\frac{3}{2}\) indicating the line falls as it moves to the right. Knowing the slope allowed us to find another point by moving 'down 3' and to the 'right 2' from the initial point \(1, -4\). This method ensures the accurate graphing of the line.