Question
Micah sketches the graph below to represent a scenario his teacher gave him.
On a graph, a line goes through points (0, 2) and (2, 4).
Which scenario could Micah’s graph represent?
Micah’s plant is 1 inch and then grows at a rate of 2 inches per week (where x is the number of weeks), for a total of y inches.
Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches.
Micah’s plant grows 2 inches per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches.
Micah’s plant grows 1 inch per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches.
On a graph, a line goes through points (0, 2) and (2, 4).
Which scenario could Micah’s graph represent?
Micah’s plant is 1 inch and then grows at a rate of 2 inches per week (where x is the number of weeks), for a total of y inches.
Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches.
Micah’s plant grows 2 inches per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches.
Micah’s plant grows 1 inch per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine which scenario Micah's graph could represent, we need to identify the equation of the line that passes through the points (0, 2) and (2, 4).
1. **Finding the slope** of the line:
The slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1
\]
2. **Finding the y-intercept**:
The line passes through the point (0, 2). This indicates that the y-intercept \( b = 2 \).
3. **Writing the equation of the line**:
The equation of the line in slope-intercept form \( y = mx + b \):
\[
y = 1x + 2 \quad \text{or} \quad y = x + 2
\]
From the equation \( y = x + 2 \):
- When \( x = 0 \) (at the start, or week 0), \( y = 2 \), which indicates the starting height of the plant is 2 inches.
- For \( x = 2 \) (after 2 weeks), \( y = 4 \), which indicates the height of the plant is 4 inches.
Now, let's review the scenarios:
1. **Micah’s plant is 1 inch and then grows at a rate of 2 inches per week (where x is the number of weeks), for a total of y inches**.
- This would correspond to the equation \( y = 2x + 1 \) (initial height of 1 inch).
- Not a match.
2. **Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches**.
- This corresponds to the equation \( y = 1x + 2 \).
- This is a match!
3. **Micah’s plant grows 2 inches per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches**.
- This would correspond to the equation \( y = 2x + 1 \).
- Not a match.
4. **Micah’s plant grows 1 inch per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches**.
- This would correspond to the equation \( y = 1x + 1 \).
- Not a match.
Thus, the scenario that fits the graph is:
**Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches.**
1. **Finding the slope** of the line:
The slope \( m \) is given by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - 0} = \frac{2}{2} = 1
\]
2. **Finding the y-intercept**:
The line passes through the point (0, 2). This indicates that the y-intercept \( b = 2 \).
3. **Writing the equation of the line**:
The equation of the line in slope-intercept form \( y = mx + b \):
\[
y = 1x + 2 \quad \text{or} \quad y = x + 2
\]
From the equation \( y = x + 2 \):
- When \( x = 0 \) (at the start, or week 0), \( y = 2 \), which indicates the starting height of the plant is 2 inches.
- For \( x = 2 \) (after 2 weeks), \( y = 4 \), which indicates the height of the plant is 4 inches.
Now, let's review the scenarios:
1. **Micah’s plant is 1 inch and then grows at a rate of 2 inches per week (where x is the number of weeks), for a total of y inches**.
- This would correspond to the equation \( y = 2x + 1 \) (initial height of 1 inch).
- Not a match.
2. **Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches**.
- This corresponds to the equation \( y = 1x + 2 \).
- This is a match!
3. **Micah’s plant grows 2 inches per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches**.
- This would correspond to the equation \( y = 2x + 1 \).
- Not a match.
4. **Micah’s plant grows 1 inch per week (where x is the number of weeks) and has a starting height of 1 inch, for a total of y inches**.
- This would correspond to the equation \( y = 1x + 1 \).
- Not a match.
Thus, the scenario that fits the graph is:
**Micah’s plant is 2 inches tall and grows at a rate of 1 inch per week (where x is the number of weeks), for a total of y inches.**
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