Question

Interpret this linear function. F(×)=3.20×, where × is amount of gas in gallons and f(×) is the price in dollars. Show work

Answer 1

To interpret the linear function \( F(x) = 3.20x \), where \( x \) represents the amount of gas in gallons and \( F(x) \) represents the price in dollars, we can break it down as follows:

1. **Understanding the Function**:
- In the equation \( F(x) = 3.20x \):
- \( F(x) \) is the dependent variable (the outcome we want to find, which is the price).
- \( x \) is the independent variable (the input, which is the amount of gas in gallons).
- The coefficient \( 3.20 \) represents the price per gallon of gas.

2. **Interpreting the Coefficient**:
- The \( 3.20 \) in front of \( x \) indicates that for every gallon of gas purchased, the cost is \$3.20. This means the price is directly proportional to the number of gallons of gas.

3. **Calculating Examples**:
- If you want to find out how much it will cost for different amounts of gas, you can substitute values of \( x \):
- For \( x = 1 \) gallon:
\[
F(1) = 3.20 \times 1 = 3.20 \text{ dollars}
\]
- For \( x = 5 \) gallons:
\[
F(5) = 3.20 \times 5 = 16.00 \text{ dollars}
\]
- For \( x = 10 \) gallons:
\[
F(10) = 3.20 \times 10 = 32.00 \text{ dollars}
\]

4. **Graphical Representation**:
- If you were to graph this linear function, the \( y \)-intercept (where the line crosses the y-axis) would be at the origin \((0, 0)\), indicating that if no gas is purchased (0 gallons), the cost is \$0.
- The slope of the line is \( 3.20 \), meaning for every additional gallon of gas, the cost increases by \$3.20.

5. **Summary**:
- The function \( F(x) = 3.20x \) represents a linear relationship between the amount of gas purchased (in gallons) and the total price (in dollars). The total price increases linearly with the quantity of gas, where each gallon costs \$3.20.