Question 57·Hard·Linear Functions
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?
For graph interpretation questions with linear models, first label clearly what each axis represents (usually input on the horizontal, output on the vertical). Then translate graph features into words: the vertical intercept is the output when the input is , and the slope is the change in output per unit change in input. Finally, match that verbal meaning—not numbers—to the answer choice that best describes it in context.
Hints
Connect the graph to the situation
Think about what the horizontal and vertical axes represent in this graph of .
Focus on the intercept
The -intercept is where the graph crosses the vertical axis. What must the weight be at that point?
Interpret that special point
At the -intercept, you know the weight and the cost. What does that say about the cost of shipping a package with that weight?
Compare to the choices
Look for the choice that describes the cost when the package weight is pounds, written in words.
Desmos Guide
Plot the given points
Enter the two data points as ordered pairs on Desmos: (2, 8.3) and (12, 20.3). You will see the two points that the cost function line must pass through.
Find the equation of the line
Use Desmos's regression feature by entering y1 ~ m x1 + b. Desmos will show values for m and b, giving you an equation of the form that fits the two points.
Identify the vertical-axis intercept
In the equation , the value of is the -intercept, because it is the value of when . This is the point where the line crosses the -axis.
Interpret that intercept in words
Think about what it means in this context for the cost to have that value when pounds. Then choose the option that describes this idea in words (a cost at zero weight).
Step-by-step Explanation
Understand what is on each axis
The function is , where is the weight (in pounds) and is the cost (in dollars).
- The horizontal axis shows (pounds).
- The vertical axis shows (dollars). So a point on the graph tells you the cost to ship a package of weight pounds.
Recall what a vertical-axis intercept means
The -intercept is where the graph crosses the -axis (the vertical axis).
- On the vertical axis, .
- So the -intercept has coordinates . This is the cost value when the weight is pounds.
Interpret in the shipping context
is the cost to ship a package that weighs pounds. In real life, this represents a starting fee or base amount that is charged even before adding any cost that depends on the weight. In other words, it is a constant dollar amount added to every shipment, regardless of how many pounds are being shipped.
Match this interpretation to the answer choices
The description that matches "the cost when " or the base amount charged regardless of weight is:
D) The fixed charge, in dollars, of shipping a package before any per-pound cost is added.