Problem 71 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. Label the x-intercept and the y-intercept. \(y=-x+8\)

Step-by-Step Solution

Answer

The x-intercept of the graph of the equation \(y = -x + 8\) is at the point (8, 0) and the y-intercept is at the point (0, 8). Also, the graph will pass through the points (-2,10) and (2,6). In general, any point (x, y) on the line will satisfy the given equation.

Step 1: Find the y-intercept

In the equation \(y = -x + 8\), the y-intercept is the constant term, which is 8. This means the line intercepts the y-axis at the point (0, 8).

Step 2: Find the x-intercept

The x-intercept can be found by setting y=0 and solving for x in the equation, which gives: \[0 = -x + 8 \]\[x = 8 \]So the x-intercept is at the point (8, 0).

Step 3: Find additional points

To draw a line, we should find more points that lie on the line. Let's select a few other values for x and find the corresponding values for y. For example:When x=-2, \(y = -(-2) + 8 = 2 + 8 = 10\), so we have the point (-2, 10);When x=2, \(y = -(2) + 8 = -2 + 8 = 6\), so we have the point (2, 6);

Step 4: Plot the graph

Now we can plot the graph using the points we found: (0,8), (8,0), (-2,10) and (2,6). We draw the line through these points and label the intercepts.

Key Concepts

Understanding the x-intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis. This particular point has a y-coordinate of zero since it lies right on the x-axis. To find the x-intercept in any linear equation, simply set y to zero and solve for x.

For example, in the equation \( y = -x + 8 \), substitute y with 0:

• \( 0 = -x + 8 \)
• Solving for x gives \( x = 8 \)

Thus, the x-intercept is the coordinate point (8, 0). It is crucial to identify this point, as it provides valuable information about the behavior of the line across the x-axis. This is very helpful when graphing because knowing where the line crosses the axes helps in sketching the linear plot accurately.

Understanding the y-intercept

The y-intercept of a line is the point where the line intersects the y-axis. This happens where x equals zero. In the slope-intercept form of a linear equation, \( y = mx + b \), the constant term \( b \) is the y-intercept.

In the provided equation \( y = -x + 8 \), we can directly observe that the y-intercept is 8. Therefore, the y-coordinate of the line at this intersection with the y-axis is 8, and the point is (0, 8).

Identifying the y-intercept is straightforward since you merely look at the constant term in the equation. This intercept is where the line exits or enters the y-axis on the graph, and it is generally the starting point when you begin plotting a linear equation on a cartesian plane.

Using a table of values for graphing

A table of values is an effective method for graphing linear equations, especially when you need to plot multiple points to ensure accuracy. You select various x-values and substitute them into the equation to find the corresponding y-values.

For the equation \( y = -x + 8 \), let's try some values:

• For \( x = -2 \), \( y = -(-2) + 8 = 10 \) gives the point (-2, 10).
• For \( x = 2 \), \( y = -(2) + 8 = 6 \) gives the point (2, 6).

By plotting these points along with the intercepts (0, 8) and (8, 0), you can draw a precise line through the points on graph paper.

Using a table of values provides multiple points through which the line can be drawn, confirming the consistent linearity of the graph and ensuring it accurately represents the equation.