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Chapter 4: Problem 40

Graph the line that has the given intercepts. \(x\) -intercept: \(-7\) \(y\) -intercept: 15

Short Answer

Expert verified
The line passing through points (-7,0) and (0,15) represents the equation with the given intercepts.

Step by step solution

01

Understand the coordinates of intercepts

Intercepts are points where the line crosses the axes. The x-intercept is given as -7, which means the line crosses the x-axis at point (-7,0). Similarly, the y-intercept is given as 15, denoting that the line crosses the y-axis at point (0,15).
02

Plot the intercepts

On an x-y graph, plot the two points, (-7,0) and (0,15). The first number in each pair is the x-coordinate and the second number is the y-coordinate. So, the point (-7,0) is 7 units to the left of the origin (0,0) on the x-axis, and the point (0,15) is 15 units above the origin on the y-axis.
03

Draw the line

After the points have been plotted, draw a straight line that goes through both of these points. This line represents the equation with the given intercepts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intercepts
Intercepts are key points on a graph where a line crosses the axes. They help define the position and direction of a line on a graph. There are two types of intercepts: x-intercepts and y-intercepts. Each plays a crucial role in graphing linear equations.
- **X-intercept**: This is where the line crosses the x-axis. The y-value is always 0 at this point.
- **Y-intercept**: This is where the line crosses the y-axis. The x-value is always 0 here.
Understanding intercepts can aid in quickly sketching the graph of a line, as they give specific solutions to the equation where one variable is zero.
x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of y is zero. Knowing the x-intercept is useful, especially when you want to graph a line quickly.

For example, an x-intercept of -7 means the point (-7,0) needs to be placed on the x-axis, seven units to the left of the origin. The x-intercept provides a starting point for drawing the line because it anchors the line on the horizontal axis. Whenever you have this intercept, it essentially marks where your line touches or crosses the horizontal line of your graph.
y-intercept
A y-intercept is the point where the line intersects the y-axis, with the x-value being zero. This point is vital in graphing because it gives a precise location where the line crosses the vertical axis.

If you have a y-intercept of 15, this represents the point (0,15). This means on the graph, the line will meet the y-axis at a height of 15 units from the origin. Plotting the y-intercept places a marker on the vertical axis, providing a key reference point that dictates the line's vertical rise. Given both intercepts, you can easily draw a line through these two strategic points.
Plotting Points
Plotting points on a graph involves defining exact coordinates on the x and y axes. These coordinates are marked on a graph by identifying the horizontal (x) and vertical (y) values. When plotting the intercepts, you can succinctly prepare to draw the entire line.

Begin by marking each intercept on the graph. For the given intercepts (-7,0) and (0,15):
  • Find -7 on the x-axis and move directly up (or down) to plot the point where y remains 0.
  • Locate 15 on the y-axis and align it horizontally where x is still 0, plotting the point.
Next, draw a line through these two points with a straight edge. This line represents your equation, as the plotted points guide you to visualize the relationship and slope of the graph.

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