Two input-output facts for a linear function, whose graph is a line, give two points on that line. In , is the slope, or change in output for each unit of input, and is the output at . Use a Desmos list regression to fit both conditions, then match the coefficients. Reversing the input and output changes gives the wrong slope.
Hints
- Hint 1
Function notation means that an input of gives an output of , making the point . What point does the other function value give you? Keep both points: one alone cannot determine a line.
- Hint 2
A regression is a Desmos fit. Write the line as , then fit its two outputs at once to the given values. Which numbers does Desmos report for , the slope, and , the output at ?
Step-by-step
Fit a line to both function values
Step 1Turn the function values into points
means the line passes through , and gives . These are two distinct points. Keep both: many lines can pass through either point alone.
- Step 2
Fit both points in Desmos
Type , where is the slope and is the -intercept, the output at . Then type . The tilde tells Desmos to fit the values; both outputs must fit the same line. Under PARAMETERS, Desmos reports and .
- Step 3
Show the coefficients as fractions
Type and on separate lines and tap each fraction button. Desmos shows and , in the form used by the choices.
- Step 4
Write the matching function
Put the negative slope in front of and the intercept after it:
This is the equation that fits both given function values. Choice D.