A home store chain is prepared to buy 3,800 of your Frog & Friends(TM) shower curtains per month for $4 each, but only 3,650 shower curtains per month for $9 each. What is the linear demand function for your Frog & Friends shower curtains?

To find the linear demand function for the Frog & Friends™ shower curtains, we need to express the relationship between the price $ p $ of the shower curtains and the quantity demanded $ q $.

### Step 1: Identify the points on the demand curve
We are given two points:
- When the price is \$4, the quantity demanded is 3,800 shower curtains. This gives the point $(q_1, p_1) = (3800, 4)$.
- When the price is \$9, the quantity demanded is 3,650 shower curtains. This gives the point $(q_2, p_2) = (3650, 9)$.

### Step 2: Determine the slope of the demand curve
The demand function is linear, so it follows the form:
\[
p = mq + b
\]
Where:
- $ m $ is the slope of the line.
- $ b $ is the y-intercept (price when quantity is zero).

To find $ m $, we use the formula for the slope between two points:
\[
m = \frac{p_2 - p_1}{q_2 - q_1} = \frac{9 - 4}{3650 - 3800} = \frac{5}{-150} = -\frac{1}{30}
\]

### Step 3: Write the equation of the line
Now that we have the slope $ m = -\frac{1}{30} $, we can substitute one of the points (say, $ q_1 = 3800 $, $ p_1 = 4 $) into the equation $ p = mq + b $ to find $ b $:
\[
4 = -\frac{1}{30}(3800) + b
\]
\[
4 = -\frac{3800}{30} + b
\]
\[
4 = -126.67 + b
\]
\[
b = 4 + 126.67 = 130.67
\]

### Step 4: Write the final demand function
The linear demand function is:
\[
p = -\frac{1}{30}q + 130.67
\]
This equation represents the price $ p $ as a function of the quantity demanded $ q $.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What happens to the demand if the price is increased to \$12?
2. How would the demand curve shift if there was a new competitor in the market?
3. Can you interpret what the y-intercept $ b = 130.67 $ means in practical terms?
4. How would a change in consumer preferences affect the linearity of this demand curve?
5. What is the price elasticity of demand based on this function?

**Tip**: Linear demand curves assume constant marginal effects; in real markets, elasticity often changes with price.

Discover how to calculate the linear demand function for Frog & Friends™ shower curtains, based on price and quantity data. This detailed explanation includes step-by-step instructions on determining the slope and writing the linear equation. Ideal for students in Grades 10-12 learning about demand functions and algebraic linear equations.

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