Question
Find the constant of variation k for the direct variation.
x f ( x )
0 0
3 6
4 8
7 14
k = –2
k = 2.5
k = 0.5
x f ( x )
0 0
3 6
4 8
7 14
k = –2
k = 2.5
k = 0.5
Answers
GPT-4o mini
In a direct variation, the relationship between \( x \) and \( f(x) \) can be expressed as \( f(x) = kx \), where \( k \) is the constant of variation.
To find \( k \), we can use any of the points provided in the table. The simplest points to calculate with are when \( x = 3 \) and \( f(x) = 6 \).
Using the formula:
\[
f(x) = kx
\]
Substituting the values:
\[
6 = k \cdot 3
\]
To find \( k \), divide both sides by 3:
\[
k = \frac{6}{3} = 2
\]
Now let's verify this with the other points to ensure consistency.
1. For \( x = 4 \) and \( f(x) = 8 \):
\[
8 = k \cdot 4 \implies k = \frac{8}{4} = 2
\]
2. For \( x = 7 \) and \( f(x) = 14 \):
\[
14 = k \cdot 7 \implies k = \frac{14}{7} = 2
\]
All points generate the same value of \( k \). Therefore, the constant of variation is:
\[
\boxed{2}
\]
It appears none of the provided options match \( k = 2 \). However, based on the calculations, the correct \( k \) value based on the given data is indeed 2.
To find \( k \), we can use any of the points provided in the table. The simplest points to calculate with are when \( x = 3 \) and \( f(x) = 6 \).
Using the formula:
\[
f(x) = kx
\]
Substituting the values:
\[
6 = k \cdot 3
\]
To find \( k \), divide both sides by 3:
\[
k = \frac{6}{3} = 2
\]
Now let's verify this with the other points to ensure consistency.
1. For \( x = 4 \) and \( f(x) = 8 \):
\[
8 = k \cdot 4 \implies k = \frac{8}{4} = 2
\]
2. For \( x = 7 \) and \( f(x) = 14 \):
\[
14 = k \cdot 7 \implies k = \frac{14}{7} = 2
\]
All points generate the same value of \( k \). Therefore, the constant of variation is:
\[
\boxed{2}
\]
It appears none of the provided options match \( k = 2 \). However, based on the calculations, the correct \( k \) value based on the given data is indeed 2.