Question

the data reflects the number of days since a plant has been transplanted (x), paired with the plant height (y).
growth (inches)
number of days
$y = 0.81x + 0.07$
how can the y-intercept of the line be interpreted?
every 0.07 days the plant grows 0.81 inches.
every 0.81 days the plant grows 0.07 inches.
initially, the plant had a height of 0.07 inches.
initially, the plant had a height of 0.81 inches.

Explanation:

Step1: Recall Slope-Intercept Form

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the \(y\) - intercept (value of \(y\) when \(x = 0\)).
In the given equation \(y=0.81x + 0.07\), \(x\) represents the number of days since the plant was transplanted and \(y\) represents the plant height.

Step2: Interpret the \(y\) - intercept

When \(x = 0\), it means the number of days since transplantation is \(0\) (i.e., initially, at the time of transplantation).
Substitute \(x = 0\) into the equation \(y=0.81x + 0.07\). We get \(y=0.81(0)+0.07=0.07\).
This means that initially (when \(x = 0\), no days have passed since transplantation), the plant's height \(y\) is \(0.07\) inches.

Now let's analyze the other options:

• Option 1: "Every 0.07 days the plant grows 0.81 inches" - The slope \(m = 0.81\) represents the rate of growth (change in \(y\) per change in \(x\)), not related to 0.07 days. So this is incorrect.
• Option 2: "Every 0.81 days the plant grows 0.07 inches" - The slope is \(0.81\) (change in \(y\) per change in \(x\)), so this misinterprets the slope and is incorrect.
• Option 4: "Initially, the plant had a height of 0.81 inches" - When \(x = 0\), \(y=0.07\) not \(0.81\), so this is incorrect.

Answer:

Initially, the plant had a height of 0.07 inches (the option with this description)