Select the correct answer.

Suppose the following function is graphed:
[tex]\[ y = \frac{8}{5}x + 4 \][/tex]

On the same grid, a new function is graphed. The new function is represented by the following equation:
[tex]\[ y = -\frac{5}{8}x + 8 \][/tex]

Which of the following statements about these graphs is true?

A. The graphs intersect at [tex]\((0, 8)\)[/tex].
B. The graph of the original function is perpendicular to the graph of the new function.
C. The graph of the original function is parallel to the graph of the new function.
D. The graphs intersect at [tex]\((0, 4)\)[/tex].



Answer :

To determine which statement is true about the graphs of the given functions, we'll analyze the slopes and intersections of the lines represented by these equations:

1. Original Function:
[tex]\[ y = \frac{8}{5}x + 4 \][/tex]
This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the y-intercept. For this function:
- Slope [tex]\( m_1 = \frac{8}{5} \)[/tex]
- Y-intercept at [tex]\((0, 4)\)[/tex]

2. New Function:
[tex]\[ y = -\frac{5}{8}x + 8 \][/tex]
Similarly, this equation is also in the slope-intercept form [tex]\(y = mx + b\)[/tex]. For this function:
- Slope [tex]\( m_2 = -\frac{5}{8} \)[/tex]
- Y-intercept at [tex]\((0, 8)\)[/tex]

Let's proceed step-by-step:

Step 1: Determine if the lines are perpendicular.

To check if the lines are perpendicular, we need to see if the slopes [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] are negative reciprocals of each other. Two lines are perpendicular if [tex]\( m_1 \cdot m_2 = -1 \)[/tex].

[tex]\[ \frac{8}{5} \cdot \left(-\frac{5}{8}\right) = -1 \][/tex]

Since the product of the slopes is [tex]\(-1\)[/tex], the lines are indeed perpendicular. Therefore, statement B (The graph of the original function is perpendicular to the graph of the new function) is true.

Step 2: Determine intersections.

We need to check if the lines intersect at specific points as mentioned in the options. We substitute the coordinates into both equations and verify:

Check intersection at [tex]\((0, 8)\)[/tex]:
- Original function: [tex]\( y = \frac{8}{5}(0) + 4 = 4 \)[/tex]
- New function: [tex]\( y = -\frac{5}{8}(0) + 8 = 8 \)[/tex]

The lines do not intersect at [tex]\((0, 8)\)[/tex], eliminating statement A.

Check intersection at [tex]\((0, 4)\)[/tex]:
- Original function: [tex]\( y = \frac{8}{5}(0) + 4 = 4 \)[/tex]
- New function: [tex]\( y = -\frac{5}{8}(0) + 8 = 8 \)[/tex]

The lines do not intersect at [tex]\((0, 4)\)[/tex], eliminating statement D.

Step 3: Determine if the lines are parallel.

Parallel lines have the same slope. Since the slopes here are [tex]\(\frac{8}{5}\)[/tex] and [tex]\(-\frac{5}{8}\)[/tex], which are not equal, the lines are not parallel, eliminating statement C.

Since we have determined the correct relationship between the slopes, the correct answer is:

B. The graph of the original function is perpendicular to the graph of the new function.