Packages shipped by a certain company are priced according to a linear model. A 2-pound package costs $8.30 to ship, and a 12-pound package costs $20.30. Let be the cost, in dollars, of shipping a package that weighs pounds, and suppose is graphed in the coordinate plane with on the horizontal axis and on the vertical axis.
What is the best interpretation of the -intercept of the graph?
An intercept is where a graph meets an axis. For a vertical-axis intercept, the horizontal input is zero, so read the output in the units the problem gives. In a linear cost model, cost at zero input is different from the rate charged for each additional unit. If the question asks only what the intercept means, you don’t need to calculate it from the two prices.
Hints
- Hint 1
At a vertical-axis intercept, the horizontal coordinate is zero. Since weight is on the horizontal axis, what weight does the -intercept represent?
- Hint 2
A linear cost model can be written as a fixed charge plus a per-pound charge. If the package weighs zero pounds, what happens to the per-pound part?
Step-by-step
Read the vertical-axis intercept
Step 1Identify the zero coordinate
No Desmos needed. The question asks what the intercept means, not for its dollar value. An intercept is where a graph meets an axis. The -axis is vertical, so its horizontal coordinate is . The -intercept shows , the shipping cost at zero pounds.
- Step 2
Interpret the cost at zero pounds
A linear model for the price has the form , where is the cost per pound and is a fixed charge. Substitute :
Remove the zero per-pound cost:
The intercept is the charge left before any per-pound cost is added. So it represents the fixed shipping charge, in dollars. Choice D.