To find the correct equation that describes the function, we can examine the given points in the table:
- When \( x = 0 \), \( y = -2 \)
- When \( x = 2 \), \( y = 4 \)
- When \( x = 4 \), \( y = 10 \)
- When \( x = 6 \), \( y = 16 \)
Let's analyze the changes in \( y \) as \( x \) increases:
- From \( (0, -2) \) to \( (2, 4) \), \( y \) increases by \( 6 \) when \( x \) increases by \( 2 \).
- From \( (2, 4) \) to \( (4, 10) \), \( y \) increases by \( 6 \) when \( x \) increases by \( 2 \).
- From \( (4, 10) \) to \( (6, 16) \), \( y \) increases by \( 6 \) when \( x \) increases by \( 2 \).
This suggests a linear relationship, and it seems that \( y \) increases consistently.
Let's determine the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - (-2)}{2 - 0} = \frac{8}{2} = 4 \]
However, the difference we calculated for the specific points has to be based only on every two units increase in \( x \). Looking at the values more closely, appearances may be misleading, so let's derive an equation to describe it.
Now, we need to find the equation in the form \( f(x) = mx + b \).
To determine \( b \) (the y-intercept), we can use the point \( (0, -2) \):
At \( x = 0 \):
\[ f(0) = b = -2 \]
Looking at the difference in \( y \) as \( x \) goes from 0:
From \( (0, -2) \) to \( (2, 4) \):
The general form appears to be:
\[ y = 3x - 2 \]
To validate \( f(x) = 3x - 2 \):
- If \( x = 0 \): \( f(0) = 3(0) - 2 = -2 \)
- If \( x = 2 \): \( f(2) = 3(2) - 2 = 6 - 2 = 4 \)
- If \( x = 4 \): \( f(4) = 3(4) - 2 = 12 - 2 = 10 \)
- If \( x = 6 \): \( f(6) = 3(6) - 2 = 18 - 2 = 16 \)
All points match correctly. Therefore, the equation that describes the function in the table is:
f(x) = 3x - 2
1 answer