Question 74·Medium·Linear Functions
The cost , in dollars, to rent a carpet-cleaning machine for hours is modeled by the equation
When the equation is graphed in the -plane, which of the following best represents the meaning of the -intercept in this context?
For questions about the meaning of a y-intercept or slope, first match the equation to the form , identify which variable is on each axis, and recall that the y-intercept is the output when the input is 0 while the slope is the rate of change. Quickly translate “input = 0” or “rate per unit” back into the story’s context (time, items, distance, etc.), then choose the option that matches that interpretation rather than focusing on specific numbers like 1 hour or maximum values.
Hints
Connect the equation to the graph
Think of like , where is on the vertical axis and is on the horizontal axis. Which part of the equation corresponds to the y-intercept?
Meaning of the y-intercept
The y-intercept is the value of the vertical variable when the horizontal variable equals 0. In this context, which quantity is 0 at the y-intercept?
Put it into words
If , what does that mean in terms of renting the machine? How would you describe the cost at that moment?
Desmos Guide
Enter the equation in Desmos
In Desmos, type the equation in terms of and : y = 40x + 75. This represents the same relationship, with standing for and standing for .
Locate the y-intercept on the graph
Look for the point where the line crosses the y-axis (the vertical axis). Click that point to see its coordinates; note the -value when , and think about what that cost means in the rental situation.
Step-by-step Explanation
Identify the variables and axes
The equation is .
- is the cost in dollars, which goes on the vertical (y) axis.
- is the number of hours, which goes on the horizontal (x) axis. So in the -plane, plays the role of and plays the role of .
Recall what a y-intercept means
The y-intercept of a graph is the point where the graph crosses the y-axis.
- On the y-axis, the horizontal coordinate is 0.
- In this problem, that means the y-intercept occurs when . So the y-intercept is the value of when .
Interpret when in context
Substitute into the equation:
This is the cost when you rent the machine for 0 hours. In words, this is a fixed starting fee you pay even if you don't use the machine for any hours.
Match the meaning to the answer choice
The y-intercept represents the cost when , which is the total cost to rent the machine for 0 hours.
So the correct answer is: The total cost to rent the machine for 0 hours.