A linear function adds the same amount to the fare for every additional mile. Two known rides give two input-output pairs; fit in Desmos to find the charge per mile and the fare at zero miles. A ride’s total fare is not the zero-mile fare unless the ride is miles. Check both rides, not just one.
Hints
- Hint 1
An input-output pair matches a distance with its fare. Turn each ride into a pair , with miles first and dollars second. What two pairs does the problem give you?
- Hint 2
In slope-intercept form, , is the charge per mile and is the fare at zero miles. Fit both known fares at once to find those two numbers.
- Hint 3
The intercept is the fare when . Don’t use the fare for a ride of several miles as that starting charge; the miles have already added to it.
Step-by-step
Approach 1: Fit both rides in Desmos
Step 1Turn the rides into points
An input-output pair puts miles first and fare second. The 3-mile ride gives , and the 8-mile ride gives . These are two points on the same line.
- Step 2
Fit the rate and zero-mile fare
Use slope-intercept form , where is dollars per mile and is the fare at miles. Type , then . The asks Desmos to fit both rides at once. Under PARAMETERS, it shows and .
- Step 3
Write the fare function
Put the fitted values into the model:
The company charges $2 per mile, plus an intercept of $8 at zero miles. Choice A.
Approach 2: Find the rate, then the starting charge
Step 1Find the charge per mile
The slope is the change in fare divided by the change in miles. Subtract the rides in the same order on top and bottom:
Type in Desmos. It prints , so each additional mile adds $2.
- Step 2
Find the charge at zero miles
The 3-mile fare is $14, so put that ride into :
Type in Desmos. Under PARAMETERS, it shows . This intercept is the zero-mile fare, not the fare for the 3-mile ride.
- Step 3
Combine the rate and charge
Put the $2-per-mile slope and the $8 zero-mile fare together:
That gives the company’s total fare for miles. Choice A.