For equal input steps, a linear function adds the same amount each time, while an exponential function multiplies by the same factor. Put the table into Desmos and divide each output by the one before it. If the ratios match, you have repeated multiplication; compare that multiplier with to find the direction. Falling values alone don't mean the function is linear.
Hints
- Hint 1
The -values move in equal steps, so compare neighboring outputs. A ratio is one number divided by another. What do you get when you divide each new output by the output immediately before it?
- Hint 2
A linear pattern changes by the same amount each step. An exponential pattern uses the same multiplier each step. Which pattern do your divisions show?
- Hint 3
A multiplier below leaves less than you started with. Once you know the type of pattern, compare its multiplier with to decide whether the function increases or decreases.
Step-by-step
Compare consecutive outputs
Step 1Find the first multiplier
The -values rise by from row to row, so compare outputs from neighboring rows. Enter the table in Desmos, with in and in . Type ; the brackets select the second and first table entries. Desmos shows , so the first output change multiplies by to get .
- Step 2
Check whether the multiplier repeats
Now type to check the other two pairs. Desmos shows . All three ratios, meaning new output divided by old output, match. A constant ratio means repeated multiplication, not the same amount subtracted each time. So the pattern is exponential, even though its outputs fall.
- Step 3
Decide the direction
The multiplier is less than , so each new output is smaller than the one before it. The function is decreasing exponential. Choice D.