To find the equation of the linear function in the form \(y = mx + b\), we will first determine the slope \(m\) and the y-intercept \(b\).
- Find the slope \(m\):
The slope \(m\) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use any two points from the table. Let's use the points \((-1, 5)\) and \((0, 8)\):
\[ m = \frac{8 - 5}{0 - (-1)} = \frac{3}{1} = 3 \]
- Find the y-intercept \(b\):
To find the y-intercept \(b\), we can substitute one of the points into the equation \(y = mx + b\). We'll use the point \((0, 8)\):
\[ y = mx + b \]
Substituting \(x = 0\) and \(y = 8\):
\[ 8 = 3(0) + b \implies b = 8 \]
- Write the equation:
Now that we have both the slope \(m = 3\) and the y-intercept \(b = 8\), we can write the equation of the linear function:
\[ y = 3x + 8 \]
Therefore, the equation of the linear function that models the relationship shown in the table is:
\[ \boxed{y = 3x + 8} \]