Question

Question
Use the table to answer the question.

x
(time in years) 0 2 4
y
(height of tree in feet) 3 6 9
Interpret the y
-intercept and slope in the context of this situation. For the slope, give the most simplified rate using whole numbers.(2 points)
The y
-intercept is
feet. The slope is
feet every
year(s).

Answer 1

To interpret the y-intercept and slope for the given data:

1. **Y-Intercept**: The y-intercept occurs when \(x = 0\). From the table, when \(x = 0\) (which corresponds to the time in years), the height of the tree \(y = 3\) feet. This means that at the start (0 years), the height of the tree is 3 feet.

**Y-Intercept Interpretation**: The y-intercept is 3 feet. This means that the initial height of the tree is 3 feet at the beginning of the observation (year 0).

2. **Slope**: The slope represents the rate of change of the height of the tree with respect to time. To find the slope, we can take two points from the table:
- Point 1: \((0, 3)\)
- Point 2: \((2, 6)\)

The slope (m) can be calculated using the formula:
\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points (2, 6) and (0, 3):
\[
m = \frac{6 - 3}{2 - 0} = \frac{3}{2}
\]

This means the tree grows 1.5 feet per 1 year. To express this as a simplified whole number rate of growth, we multiply the numerator and denominator by 2 to avoid fractions:

\[
\text{Simplified Rate} = \frac{3 \text{ feet}}{2 \text{ years}} \implies 3 \text{ feet every } 2 \text{ year(s)}
\]

**Slope Interpretation**: The slope is 3 feet every 2 years, which indicates that the height of the tree increases by 3 feet for every 2 years that pass.

In summary:
- The y-intercept is **3 feet**.
- The slope is **3 feet every 2 years**.