To find the slope \( m \) of the line that represents the relationship between the number of years and the height of Destiny's tree, we can use the formula for slope:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we have two points:
- Point 1 (after 2 years): \( (2, 80) \)
- Point 2 (after 8 years): \( (8, 170) \)
Let’s label the points:
- \( (x_1, y_1) = (2, 80) \)
- \( (x_2, y_2) = (8, 170) \)
Now we substitute these values into the slope formula:
\[ m = \frac{170 - 80}{8 - 2} \] \[ m = \frac{90}{6} = 15 \]
So, the slope \( m \) is \( 15 \). This means the tree grows 15 inches per year.
Next, we need to write the equation of the line in the form \( y = mx + b \). We already have \( m = 15 \).
To find \( b \) (the y-intercept), we can use one of the points. We'll use the point \( (2, 80) \):
\[ y = mx + b \implies 80 = 15(2) + b \] \[ 80 = 30 + b \] \[ b = 80 - 30 = 50 \]
The y-intercept \( b \) is \( 50 \).
Now we can write the equation of the line that represents the height of the tree over time:
\[ y = 15x + 50 \]
Where:
- \( y \) is the height in inches,
- \( x \) is the number of years since planting.
Summary
- Slope \( (m) = 15 \)
- Y-Intercept \( (b) = 50 \)
- Equation: \( y = 15x + 50 \)
Plotting Points
To plot the points on the graph based on the data:
- Point (2, 80): From the origin, move 2 units right (for 2 years) and 80 units up (for 80 inches).
- Point (8, 170): Move 8 units right (for 8 years) and 170 units up (for 170 inches).
Lastly, draw a line through these points to represent the growth of the tree over the years.