You buy a raspberry bush that you plant. In the first month after planting, the raspberry bush grows the same amount every day. 6 days after planting, the raspberry bush is 0.50 meters high. 20 days after planting, the raspberry bush is 0.92 meters high. Set up a function that shows how the raspberry bush's total height y, in meters, depends on how many days have passed since planting.

To set up a function that describes the height \( y \) of the raspberry bush as a function of the number of days \( x \) that have passed since planting, we'll use the information given.

### Information Given:
- After 6 days (\( x = 6 \)), the height of the bush is 0.50 meters (\( y = 0.50 \)).
- After 20 days (\( x = 20 \)), the height of the bush is 0.92 meters (\( y = 0.92 \)).

### Step 1: Determine the Slope \( m \)

The height \( y \) increases linearly with the number of days \( x \), so we can use the slope formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values:

\[
m = \frac{0.92 - 0.50}{20 - 6} = \frac{0.42}{14} = 0.03
\]

So, the slope \( m \) is \( 0.03 \) meters per day.

### Step 2: Determine the y-intercept \( b \)

Using the point \( (6, 0.50) \) and the slope \( m = 0.03 \), we can find the y-intercept \( b \) using the equation:

\[
y = mx + b
\]

Substitute \( x = 6 \) and \( y = 0.50 \):

\[
0.50 = 0.03(6) + b
\]

\[
0.50 = 0.18 + b \\
b = 0.50 - 0.18 = 0.32
\]

### Final Function

The function that shows the height \( y \) of the raspberry bush in meters as a function of the number of days \( x \) since planting is:

\[
y = 0.03x + 0.32
\]

This equation means that the raspberry bush grows at a rate of 0.03 meters per day, and it started with a height of 0.32 meters on the day it was planted.

Would you like more details, or do you have any questions?

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Here are some related questions:
1. How tall will the raspberry bush be after 10 days?
2. What would the height be after 0 days (at planting)?
3. How many days will it take for the raspberry bush to reach a height of 1.2 meters?
4. If the growth rate changed to 0.04 meters per day, what would the new equation be?
5. What happens to the function if the bush starts at 0.20 meters instead of 0.32 meters?
6. Why is it important to know the slope in this problem?
7. How does the y-intercept affect the interpretation of the bush's initial height?
8. Can this function predict the height of the bush beyond 30 days accurately?

**Tip:** When analyzing linear growth problems, always check that the rate of change (slope) makes sense in the context of the problem.

Explore how to calculate the height of a raspberry bush over time using a linear function. Understand the growth rate and initial height from specific data points at different days after planting. Suitable for students in Grades 7-9.

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