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 (0,8).
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 (0,4).
Answer (1)
The y-intercept of y = 5 8 x + 4 is 4, and the y-intercept of y = − 8 5 x + 8 is 8.
The slope of y = 5 8 x + 4 is 5 8 , and the slope of y = − 8 5 x + 8 is − 8 5 .
The product of the slopes is 5 8 × − 8 5 = − 1 .
Since the product of the slopes is -1, the lines are perpendicular: $\boxed{B}.
Explanation
Understanding the Problem We are given two linear equations: y = 5 8 x + 4 and y = − 8 5 x + 8 . We need to determine which of the given statements is true about the graphs of these two lines.
Analyzing the Statements Let's analyze each statement:
A. The graphs intersect at ( 0 , 8 ) :
For the first equation, if x = 0 , then y = 5 8 ( 0 ) + 4 = 4 . So, the first line intersects the y-axis at ( 0 , 4 ) .
For the second equation, if x = 0 , then y = − 8 5 ( 0 ) + 8 = 8 . So, the second line intersects the y-axis at ( 0 , 8 ) .
Since the y-intercepts are different, the lines do not intersect at ( 0 , 8 ) .
D. The graphs intersect at ( 0 , 4 ) :
As we found above, the first line intersects the y-axis at ( 0 , 4 ) , but the second line intersects the y-axis at ( 0 , 8 ) . Therefore, the lines do not intersect at ( 0 , 4 ) .
C. The graph of the original function is parallel to the graph of the new function:
Parallel lines have the same slope. The slope of the first line is 5 8 , and the slope of the second line is − 8 5 . Since the slopes are different, the lines are not parallel.
B. The graph of the original function is perpendicular to the graph of the new function:
Perpendicular lines have slopes that are negative reciprocals of each other. In other words, the product of their slopes is -1. The slope of the first line is 5 8 , and the slope of the second line is − 8 5 . Let's multiply the slopes: 5 8 × − 8 5 = − 1 . Since the product of the slopes is -1, the lines are perpendicular.
Conclusion Since the product of the slopes of the two lines is -1, the graphs of the two functions are perpendicular. Therefore, statement B is true.
Examples
Understanding the relationships between lines, such as perpendicularity, is crucial in various real-world applications. For instance, architects and engineers use these concepts to design buildings and structures, ensuring that walls are perpendicular to the ground for stability. In navigation, understanding perpendicular paths helps in determining the shortest distance between two points or in designing road layouts that minimize traffic congestion. These mathematical principles provide a foundation for creating safe and efficient designs in numerous fields.
