Question

from unit 3, lesson 4
a seed is planted in a glass pot, and its height is measured in centimeters every day.
the best - fit line is given by the equation ( y = 0.404x - 5.18 ), where ( y ) represents the height of the plant above ground level, and ( x ) represents the number of days since it was planted.
a. what is the slope of the best - fit line?
what does the slope of the line mean in this situation?
is it reasonable?
b. what is the ( y ) - intercept of the best - fit line?
what does the ( y ) - intercept of the line mean in this situation?
is it reasonable?

Explanation:

Step1: Identify the slope from the linear equation

The equation of a line is in the form \(y = mx + b\), where \(m\) is the slope. For the equation \(y=0.404x - 5.18\), the slope \(m = 0.404\).
The slope represents the rate of change. In this context, it means that the height of the plant increases by \(0.404\) centimeters per day. This is reasonable as it shows a positive growth rate for the plant over time.

Step2: Identify the y - intercept from the linear equation

In the equation \(y = mx + b\), \(b\) is the y - intercept. For \(y=0.404x - 5.18\), the y - intercept \(b=- 5.18\).
The y - intercept represents the value of \(y\) when \(x = 0\). In this situation, it would mean the height of the plant \(0\) days after being planted (at the time of planting). A negative value might seem odd at first glance, but it could imply that the measurement is relative to the ground level and perhaps the seed was planted slightly below the measurement reference point (e.g., if the ground level is at \(y = 0\) and the seed was planted \(5.18\) cm below that initial reference for measurement purposes). So, it can be considered reasonable within the context of the measurement setup.

Answer:

a. The slope of the best - fit line is \(0.404\). It means the plant's height increases by \(0.404\) cm per day. It is reasonable.
b. The y - intercept of the best - fit line is \(-5.18\). It means the height of the plant \(0\) days after being planted. It is reasonable (assuming a measurement reference point where the seed is below ground level at planting).