Question

Which of the following equations correctly describes the function in the table using function notation?

x y
0 -2
2 4
4 10
6 16


(1 point)
Responses

f(x)=2x
f of x is equal to 2 x

f(x)=2x+4
f of x is equal to 2 x plus 4

f(x)=x−2
f of x is equal to x minus 2

f(x)=3x−2

Answer 1

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**