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Sagot :
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.
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.
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