To find the equation of the linear function in the form \(y = mx + b\), we need to determine the slope \(m\) and the y-intercept \(b\) using the given points \((6, 16)\) and \((15, 43)\).
Step 1: Calculate the slope \(m\)
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our case, \((x_1, y_1) = (6, 16)\) and \((x_2, y_2) = (15, 43)\).
Plugging in the values:
\[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]
Step 2: Use the slope-point form to find \(b\)
Now, we can use the slope and one of the points to solve for \(b\). We'll use the point \((6, 16)\).
The equation in slope-intercept form is:
\[ y = mx + b \]
Substituting \(m = 3\), \(x = 6\), and \(y = 16\):
\[ 16 = 3(6) + b \] \[ 16 = 18 + b \]
Now, solve for \(b\):
\[ b = 16 - 18 = -2 \]
Step 3: Write the final equation
Now we have both \(m\) and \(b\):
\[ y = 3x - 2 \]
Thus, the equation of the linear function that models the relationship shown in the table is:
\[ \boxed{y = 3x - 2} \]