An online clothing company sells custom sweatshirts. The company charges $$ \$ 6.50 $$ for each sweatshirt and a flat fee of $$ \$ 3.99 $$ for shipping. a. Write a linear function in the form $$ y=m x+b $$ that represents the total cost, $$ y $$, in dollars, for a single order of $$ x $$ sweatshirts. b. Describe how the linear function would change if the shipping charge applied to each sweatshirt.

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Let's break down the problem step by step.

### Part a: Writing the Linear Function

1. **Identify the components of the cost:**
   - The cost per sweatshirt is $$ \$ 6.50 $$.
   - The flat shipping fee is $$ \$ 3.99 $$.

2. **Define the variables:**
   - Let $$ x $$ be the number of sweatshirts ordered.
   - Let $$ y $$ be the total cost in dollars.

3. **Formulate the linear function:**
   - The total cost $$ y $$ can be expressed as the sum of the cost of the sweatshirts and the shipping fee.
   - The cost of $$ x $$ sweatshirts is $$ 6.50x $$.
   - The total cost function can be written as:
     $$
     y = 6.50x + 3.99
     $$

Thus, the linear function in the form $$ y = mx + b $$ is:
$$
y = 6.50x + 3.99
$$
where $$ m = 6.50 $$ (the slope, representing the cost per sweatshirt) and $$ b = 3.99 $$ (the y-intercept, representing the flat shipping fee).

### Part b: Change in the Linear Function with Shipping Charge per Sweatshirt

If the shipping charge applied to each sweatshirt, the cost structure would change as follows:

1. **New cost components:**
   - The cost per sweatshirt would now include the shipping fee. If the shipping fee is $$ \$ 3.99 $$ for each sweatshirt, the new cost per sweatshirt would be:
     $$
     6.50 + 3.99 = 10.49
     $$

2. **New linear function:**
   - The total cost $$ y $$ for $$ x $$ sweatshirts would now be:
     $$
     y = 10.49x
     $$

In this case, the linear function would be:
$$
y = 10.49x
$$
where there is no flat shipping fee, and the cost per sweatshirt includes the shipping charge.

### Summary
- The original function is $$ y = 6.50x + 3.99 $$.
- If the shipping charge applies to each sweatshirt, the new function is $$ y = 10.49x $$.
a. The total cost $$ y $$ in dollars for $$ x $$ sweatshirts is $$ y = 6.50x + 3.99 $$. b. If the shipping charge applies to each sweatshirt, the total cost becomes $$ y = 10.49x $$.

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