Problem 70 - College Algebra with Corequisite Support

Question

For the following exercises, sketch a line with the given features. An \(x\) -intercept of (-4,0) and \(y\) -intercept of (0,-2)

Step-by-Step Solution

Answer

Plot the intercepts and draw a line through them.

Step 1: Plotting the Intercepts

First, identify and plot the given intercepts on the coordinate plane. The x-intercept is (-4,0) which means this point lies on the x-axis. Similarly, the y-intercept is (0,-2) which lies on the y-axis. Mark both points on the graph.

Step 2: Drawing the Line

Now that both intercepts are plotted on the graph, draw a straight line passing through these two points. Use a ruler for precision if necessary to ensure the line is straight.

Step 3: Double-checking the Line

Verify that the drawn line accurately passes through both intercept points. If it does, then the task is complete. Ensure that the line extends infinitely in both directions beyond the intercepts, as lines stretch infinitely in geometry.

Key Concepts

x-intercept

In coordinate geometry, the x-intercept is a critical point where a graph intersects the x-axis. At this point, the y-coordinate is always zero because it lies on the horizontal axis where there is no vertical displacement. In the given problem, the x-intercept is

• (-4, 0)

This tells you that this particular graph crosses the x-axis at the point where x equals -4, and y equals 0.
The x-intercept is important because it provides you with one point of reference to draw your line or curve on a graph. To find the x-intercept of any equation, set the y variable to zero and solve for x. This will give you the exact location where the graph hits the x-axis.

y-intercept

Understanding the y-intercept is key in graphing linear equations. The y-intercept is where the graph crosses the y-axis. At this intersecting point, the x-coordinate will always be zero because it is positioned on the vertical axis. For the line given in the problem, the y-intercept is

• (0, -2)

This means the line crosses the y-axis at the point where y equals -2.
The y-intercept offers another critical point needed to define the line on a coordinate plane. To locate the y-intercept from an equation, set x to zero and solve for y. This calculation provides the specific y-coordinate at which the graph intersects the y-axis. Knowing the y-intercept helps in sketching the graph quickly and accurately.

graph plotting

Graph plotting involves marking points on a coordinate plane and drawing the line or curve that connects them. In coordinate geometry, plotting is crucial for visually representing equations and understanding the relationships between variables. The first step in graph plotting is to accurately mark the intercepts on the graph.

• Plot the x-intercept: (-4, 0)
• Plot the y-intercept: (0, -2)

Once these points are plotted, the next step is to connect them with a straight line. A ruler can help ensure precision while drawing to avoid any errors.
After plotting, the drawn line should be a true representation of the linear relationship. It has to extend infinitely in both directions, symbolizing how lines continue endlessly. This step not only helps to visually express the equation but also verifies the solution.

linear equations

Linear equations form the foundation of coordinate geometry. They describe relationships between variables in the form of a straight line when graphed. A typical linear equation can be written in the form of

• y = mx + b

where *m* denotes the slope of the line and *b* represents the y-intercept.
The y-intercept is the point where the line crosses the y-axis, and this equation format makes it easier to plot the graph.
A linear equation exhibits a constant rate of change, representing a simple one-to-one relationship between the variables. Graphically, it is expressed as a straight line which can be easily visualized by knowing just two points: the x-intercept and the y-intercept. From these points, you can draw the complete graph of the equation.