The initial value of a linear function can typically be found by looking at the point where the line intersects the y-axis, which corresponds to the value of the dependent variable when the independent variable (in this case, the number of days using the new fertilizer) is zero.
From the points given in the problem, we see that the height of the stalk of corn is increasing as the number of days increases. We also observe the points provided:
- (3, 10): 3 days → 10 feet
- (4, 12): 4 days → 12 feet
- (5, 14): 5 days → 14 feet
- (6, 16): 6 days → 16 feet
The increase in height is consistent, suggesting a linear function.
To determine the initial value (the height at 0 days), we can find the equation of the line. We can set up a slope using any two of the points. Let's use the points (3, 10) and (4, 12):
The slope (m) is calculated as follows: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 10}{4 - 3} = \frac{2}{1} = 2 \]
Now that we have the slope, we can use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \]
Using the point (3, 10): \[ y - 10 = 2(x - 3) \] \[ y - 10 = 2x - 6 \] \[ y = 2x + 4 \]
This equation, \( y = 2x + 4 \), represents the height (y) of the stalk of corn as a function of the number of days (x).
To find the initial value (height when x = 0): \[ y(0) = 2(0) + 4 = 4 \]
Thus, the initial value is 4 feet.