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