To find the equation of the linear function that models the relationship shown in the table, we need to determine the slope (m) and the y-intercept (b).
Given points from the table:
- Point 1: (6, 16)
- Point 2: (15, 43)
Step 1: Finding the slope (m)
The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in our points:
\[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]
Step 2: Finding the y-intercept (b)
Now we can use one of the points to find the y-intercept (b). We'll use the point (6, 16).
The equation of a line in slope-intercept form is:
\[ y = mx + b \]
Substituting \(m = 3\) and the point (6, 16):
\[ 16 = 3(6) + b \]
This simplifies to:
\[ 16 = 18 + b \implies b = 16 - 18 \implies b = -2 \]
Step 3: Writing the equation
Now we can write the equation of the linear function:
\[ y = 3x - 2 \]
Thus, the final answer is:
\[ y = 3x - 2 \]