Question
The graph of f(x) and table for g(x) = f(kx) are given.
The graph shows an upward opening parabola labeled f of x that passes through a point negative 2 comma 8, a point negative 1 comma 2, a vertex 0 comma 0, a point 1 comma 2, and a point 2 comma 8.
x g(x)
−10 8
−5 2
0 0
5 2
10 8
What is the value of k?
k is equal to one fifth
k = 5
k is equal to one tenth
k = 10
The graph shows an upward opening parabola labeled f of x that passes through a point negative 2 comma 8, a point negative 1 comma 2, a vertex 0 comma 0, a point 1 comma 2, and a point 2 comma 8.
x g(x)
−10 8
−5 2
0 0
5 2
10 8
What is the value of k?
k is equal to one fifth
k = 5
k is equal to one tenth
k = 10
Answers
GPT-4o mini
To determine the value of \( k \) in the function \( g(x) = f(kx) \), we need to analyze how the graph of \( f(x) \) is transformed to obtain \( g(x) \).
The given points on the graph of \( f(x) \) are:
- \( (0, 0) \) (vertex)
- \( (-1, 2) \)
- \( (1, 2) \)
- \( (-2, 8) \)
- \( (2, 8) \)
The graph of \( g(x) \) gives us the corresponding values at the following inputs:
- \( g(-10) = 8 \)
- \( g(-5) = 2 \)
- \( g(0) = 0 \)
- \( g(5) = 2 \)
- \( g(10) = 8 \)
From the transformations involved in \( g(x) = f(kx) \), we can tell that when we evaluate \( g(-10) = f(-10k) \), \( g(-5) = f(-5k) \), etc. We know that \( f(x) \) is symmetric about the y-axis and holds certain key points at certain x-values.
The point \( g(0) = 0 \) directly corresponds to the vertex of \( f(x) \) when \( kx = 0 \) which happens when \( x = 0 \). Therefore, this does not provide k directly.
Now consider:
- \( g(10) = 8 \) corresponds to some \( x \) value where \( f(x) = 8 \). We have two values in \( f(x) \) equal to \( 8 \): at \( x = -2 \) and \( x = 2 \).
Thus, to find out which \( k \) causes \( -10k = -2 \) or \( 10k = 2 \):
1. If \( 10k = 2 \), then \( k = \frac{2}{10} = \frac{1}{5} \).
2. If \( -10k = -2 \), similarly results in \( k = \frac{1}{5} \) also.
Next, check the point \( g(-5) = 2 \):
- We find if \( -5k = -1 \) gives \( k = \frac{1}{5} \); similarly this checks out.
Thus, \( k \) is confirmed to be \( \frac{1}{5} \).
The answer is:
\[
k = \frac{1}{5}
\]
The given points on the graph of \( f(x) \) are:
- \( (0, 0) \) (vertex)
- \( (-1, 2) \)
- \( (1, 2) \)
- \( (-2, 8) \)
- \( (2, 8) \)
The graph of \( g(x) \) gives us the corresponding values at the following inputs:
- \( g(-10) = 8 \)
- \( g(-5) = 2 \)
- \( g(0) = 0 \)
- \( g(5) = 2 \)
- \( g(10) = 8 \)
From the transformations involved in \( g(x) = f(kx) \), we can tell that when we evaluate \( g(-10) = f(-10k) \), \( g(-5) = f(-5k) \), etc. We know that \( f(x) \) is symmetric about the y-axis and holds certain key points at certain x-values.
The point \( g(0) = 0 \) directly corresponds to the vertex of \( f(x) \) when \( kx = 0 \) which happens when \( x = 0 \). Therefore, this does not provide k directly.
Now consider:
- \( g(10) = 8 \) corresponds to some \( x \) value where \( f(x) = 8 \). We have two values in \( f(x) \) equal to \( 8 \): at \( x = -2 \) and \( x = 2 \).
Thus, to find out which \( k \) causes \( -10k = -2 \) or \( 10k = 2 \):
1. If \( 10k = 2 \), then \( k = \frac{2}{10} = \frac{1}{5} \).
2. If \( -10k = -2 \), similarly results in \( k = \frac{1}{5} \) also.
Next, check the point \( g(-5) = 2 \):
- We find if \( -5k = -1 \) gives \( k = \frac{1}{5} \); similarly this checks out.
Thus, \( k \) is confirmed to be \( \frac{1}{5} \).
The answer is:
\[
k = \frac{1}{5}
\]