To find a function that represents the linear relationship between the number of years since the trees were planted (x) and the average height of the trees in feet (y), we can follow these steps:
-
Identify the Data Points: The pairs of (x, y) values from the table are:
- (1, 10)
- (3, 24)
- (6, 45)
- (11, 80)
- (15, 105)
-
Calculate the Slope (m): The linear relationship can be represented as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
We will calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's use the points (1, 10) and (15, 105):
\[ m = \frac{105 - 10}{15 - 1} = \frac{95}{14} \approx 6.7857 \]
-
Find the y-intercept (b): We can use one of the points and the slope to find \( b \). Using the point (1, 10):
\[ y = mx + b \] \[ 10 = 6.7857 \cdot 1 + b \] \[ b = 10 - 6.7857 \approx 3.2143 \]
-
Write the Function: Putting it all together, we have:
\[ y = 6.7857x + 3.2143 \]
To simplify the coefficients, we can keep them as fractions if converting the calculations precisely to standard form. It's also good practice to test the linearity by checking other points to confirm that they fit.
Final Function
The function that represents the situation is approximately:
\[ y \approx 6.79x + 3.21 \]
If you prefer using more exact fractions, the slope can be expressed as \( \frac{95}{14} \) and y-intercept can be numerically approximated further, but typically this straightforward approach is what we describe for calculations.