The linear function satisfies and .
Which equation defines ?
A linear function has a constant rate of change, so two different input-output pairs determine its rule. Read each function value as a point, then use a Desmos regression to fit a rule of the form . If the output falls as the input rises, the coefficient of must be negative; that direction helps catch a sign error.
Hints
- Hint 1
A function value names a point: means the graph contains . Turn each given value into an input-output pair. Which number is the input in each pair?
- Hint 2
In , is the slope, or output change per input step. Compare the two pairs: as the input increases, does the output rise or fall?
- Hint 3
A Desmos regression uses to fit unknown constants. Put the two function values in one list and their matching outputs in another list, in the same order.
Step-by-step
Fit a linear rule in Desmos
Step 1Turn the function values into points
means input gives output , so the graph contains . Likewise, gives . Keep each input paired with its output.
- Step 2
Check the direction of the line
From to , the input increases while the output drops from to . The slope is the output change per input step, so the coefficient of must be negative. Direction alone doesn't tell you the full equation.
- Step 3
Fit both values and write the rule
A linear rule has the form . Type , then . The tells Desmos to fit both outputs at once; list positions keep the pairs matched. Under PARAMETERS, Desmos shows and . Here is the slope and is the intercept, the output when . Two distinct input-output pairs pin down one line. So the function is . Choice B.