Micah sketches the graph below to represent a scenario his teacher gave him.

Question

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.

Answer 1

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 $y=mx+b$: $$y=1x+2\text{or}y=x+2$$

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.