Question

Multiple Choice Given that the intercepts of a graph are (-4,0) and \((0,5),\) choose the statement that is true. (a) The \(y\) -intercept is \(-4,\) and the \(x\) -intercept is 5 . (b) The \(y\) -intercepts are -4 and 5 . (c) The \(x\) -intercepts are -4 and 5 . (d) The \(x\) -intercept is \(-4,\) and the \(y\) -intercept is 5 .

Short answer

Statement (d) is true: The x-intercept is -4, and the y-intercept is 5.

Step-by-step solution

Identify the Intercepts

Given the points (-4,0) and (0,5), determine which axes they intersect. The point (-4,0) lies on the x-axis, and the point (0,5) lies on the y-axis.

Define the x-intercept

An x-intercept is a point where the graph crosses the x-axis. This happens where the y-coordinate is 0. Therefore, (-4,0) is the x-intercept.

Define the y-intercept

A y-intercept is a point where the graph crosses the y-axis. This happens where the x-coordinate is 0. Therefore, (0,5) is the y-intercept.

Determine the Correct Statement

From the definitions above, we can determine that the x-intercept is -4, and the y-intercept is 5. This matches statement (d).

Key concepts

x-intercept

In mathematics, the x-intercept of a graph is the point where it crosses the x-axis. This is the location where the value of the y-coordinate is zero. To find the x-intercept, look for the point on the graph where y equals zero. For example, consider the point (-4, 0). Here, the y-coordinate is zero, so this point lies on the x-axis and is the x-intercept. It means that at x = -4, the graph intersects the x-axis, fulfilling the condition that y must be zero.

y-intercept

The y-intercept of a graph is the point where it crosses the y-axis. At this point, the x-coordinate is zero. Identifying the y-intercept involves finding where x equals zero on the graph. Let's take the point (0, 5). This point's x-coordinate is zero, signifying that it lies on the y-axis. Therefore, (0,5) is the y-intercept. This indicates that the graph intersects the y-axis at y = 5.

coordinate system

The coordinate system is a framework for plotting points on a graph. It consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, which is the point (0,0). Each point on the coordinate plane is represented by a pair of numbers (x, y). The first number, x, indicates the position along the x-axis, while the second number, y, shows the position along the y-axis. This system allows us to precisely locate points and navigate the plane.

graph analysis

Analyzing a graph involves interpreting the data it represents and understanding key points, such as intercepts. Graph analysis includes:
• Identifying intercepts: These are the points where the graph intersects the axes. The x-intercept occurs where y=0, and the y-intercept occurs where x=0.
• Understanding the graph's slope: This indicates the rate of change and direction of the line.
• Plotting points: Using the coordinate system to accurately place points.
• Interpreting relationships: Evaluating how variables relate to one another.
Each of these aspects helps in making sense of the graphical representation of data.