Question

Steven recorded the growth of a plant over 10 weeks for his science project. He made a graph that shows how much the plant grew each week. The plant was 2 in. tall when Steven started his project. At week 5, the plant was 8 in. tall. (a) What is the equation of the line, first written in point-slope form? What is the equation of the line, first written in point-slope form and then written in slope-intercept form? Show how you determined the equation. (b) Based on the linear model, approximately how much did the plant grow each week?

Answer 1

The equation of the line that models the growth of the plant can be written in point-slope form as:

y - 2 = m(x - 5)

where y is the height of the plant (in inches) at any given week x, and m is the slope of the line (which represents the rate of growth).

To find the slope m, we can use the fact that the plant was 2 inches tall at week 5, so the point (5, 2) is on the line. Using the slope formula, we have:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. In this case, (x1, y1) = (5, 2) and (x2, y2) = (10, 8), so:

m = (8 - 2) / (10 - 5) = 6 / 5 = 1.2

Now we can write the equation of the line in slope-intercept form as:

y = 1.2x + 2

Step-by-step explanation:

To find the equation of the line that models the growth of the plant, we need to find the slope of the line and the y-intercept. The slope tells us the rate of growth, and the y-intercept tells us the starting height of the plant.

First, we can find the slope of the line by using the fact that the plant was 2 inches tall at week 5. Let’s call the height of the plant at week 5 “h”. Then we can write:

h = 2

Next, we can find the height of the plant at week 10 by using the fact that the plant grew by 6 inches over the 10 weeks:

h10 = h + 6

Substituting h = 2, we get:

h10 = 2 + 6 = 8

Now we can find the slope of the line by using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. In this case, (x1, y1) = (5, 2) and (x2, y2) = (10, 8), so:

m = (8 - 2) / (10 - 5) = 6 / 5 = 1.2

Finally, we can write the equation of the line in point-slope form as:

y - 2 = m(x - 5)

or in slope-intercept form as:

y = 1.2x + 2