Question

Which equation represents the relationship between X and y X negative 1 0 1 2 3 y -4 -1 2 5 8

Answer 1

To find the relationship between \( x \) and \( y \) based on the given values:

\[
\begin{align*}
x: & \quad -1, \quad 0, \quad 1, \quad 2, \quad 3 \\
y: & \quad -4, \quad -1, \quad 2, \quad 5, \quad 8
\end{align*}
\]

We can observe the values of \( y \) in relation to \( x \). Let’s calculate the differences in \( y \) as \( x \) increases:

1. From \( x = -1 \) to \( x = 0 \): \( y \) changes from \(-4\) to \(-1\) (change of \(3\))
2. From \( x = 0 \) to \( x = 1 \): \( y \) changes from \(-1\) to \(2\) (change of \(3\))
3. From \( x = 1 \) to \( x = 2 \): \( y \) changes from \(2\) to \(5\) (change of \(3\))
4. From \( x = 2 \) to \( x = 3 \): \( y \) changes from \(5\) to \(8\) (change of \(3\))

The change in \( y \) is consistent at \(3\) for each increase of \(1\) in \( x\). This suggests a linear relationship.

We can find the slope \( m \) of the line:

\[
m = \frac{\Delta y}{\Delta x} = \frac{3}{1} = 3
\]

Next, we can use point-slope form to find the equation of the line. Using the point \( (0, -1) \):

\[
y - (-1) = 3(x - 0)
\]

This simplifies to:

\[
y + 1 = 3x
\]

Subtracting \(1\) from both sides gives:

\[
y = 3x - 1
\]

Thus, the equation that represents the relationship between \( x \) and \( y \) is:

\[
\boxed{y = 3x - 1}
\]

You can verify this equation by plugging in the values of \( x \) from the table and checking that they return the corresponding \( y \) values.