Question
Dalvin bought a plant and planted it in a pot near a window in his house. Let H represent the height of the plant, in inches, t months since Dalvin bought the plant. A graph of H is shown below. Write an equation for H then state the y-intercept of the graph and determine its interpretation in the context of the problem.
Answers
GPT-4o mini
To write an equation for the height of the plant \( H \) in terms of the time \( t \) (in months), we can assume that the graph likely represents a linear relationship, given that plant growth often follows a consistent pattern over time.
If we assume that the graph shows a linear growth model, the equation can generally be expressed in the slope-intercept form:
\[ H(t) = mt + b \]
where:
- \( H(t) \) is the height of the plant at time \( t \),
- \( m \) is the rate of growth (slope) of the plant in inches per month,
- \( b \) is the initial height of the plant (y-intercept).
### Finding the Equation
If you have specific values from the graph \( (t_1, H_1) \) and \( (t_2, H_2) \), you can calculate the slope \( m \) as follows:
\[ m = \frac{H_2 - H_1}{t_2 - t_1} \]
Next, you can find the y-intercept \( b \) by determining the height of the plant when \( t = 0 \). This corresponds to the height of the plant when Dalvin bought it.
### Y-Intercept
The y-intercept \( b \) indicates the height of the plant when \( t = 0 \) (the time when Dalvin bought the plant). This value provides insight into the initial condition of the plant before any growth occurred.
### Interpretation
In the context of the problem, the y-intercept \( b \) would represent the initial height of the plant in inches at the time Dalvin purchased it. For instance, if the y-intercept is 5, this means that the plant was 5 inches tall when Dalvin bought it.
If you can provide the slope and initial height from the graph (or other specific points), I can help you formulate the complete equation.
If we assume that the graph shows a linear growth model, the equation can generally be expressed in the slope-intercept form:
\[ H(t) = mt + b \]
where:
- \( H(t) \) is the height of the plant at time \( t \),
- \( m \) is the rate of growth (slope) of the plant in inches per month,
- \( b \) is the initial height of the plant (y-intercept).
### Finding the Equation
If you have specific values from the graph \( (t_1, H_1) \) and \( (t_2, H_2) \), you can calculate the slope \( m \) as follows:
\[ m = \frac{H_2 - H_1}{t_2 - t_1} \]
Next, you can find the y-intercept \( b \) by determining the height of the plant when \( t = 0 \). This corresponds to the height of the plant when Dalvin bought it.
### Y-Intercept
The y-intercept \( b \) indicates the height of the plant when \( t = 0 \) (the time when Dalvin bought the plant). This value provides insight into the initial condition of the plant before any growth occurred.
### Interpretation
In the context of the problem, the y-intercept \( b \) would represent the initial height of the plant in inches at the time Dalvin purchased it. For instance, if the y-intercept is 5, this means that the plant was 5 inches tall when Dalvin bought it.
If you can provide the slope and initial height from the graph (or other specific points), I can help you formulate the complete equation.