An exponential function multiplies by a fixed factor over equal input intervals. Divide the later output by the earlier one to find the factor across the gap between their inputs. Then start the exponent at the earlier input and make it count whole intervals, not single units. A factor for a multi-unit interval needs an exponent divided by the interval’s length.
Hints
- Hint 1
For an exponential function, dividing the later output by the earlier one gives the factor across the interval between their inputs. How many input units pass between the two given values, and what is their output ratio?
- Hint 2
Start the exponent at zero where the known output is ; then the function will equal there. An interval is one full stretch between the given inputs. How can the exponent reach at the other input?
- Hint 3
An exponential factor below makes the output shrink. Check a candidate at both given inputs: it must match both outputs, not merely pass through the starting value.
Step-by-step
Find the four-unit factor
Step 1Find the factor across the given interval
From to is a four-unit interval. Because is exponential, its outputs multiply by a fixed factor over that interval. Type in Desmos. It shows , which equals , so the output is multiplied by every four input units.
- Step 2
Count intervals starting at 3
The given is the output at , not necessarily at . Use as the reference input, where the exponent starts at . The exponent counts four-unit intervals since then: it is at and at . So gets applied once, not four times.
- Step 3
Write and check the function
Type in Desmos, then type and . Desmos prints and . The starting output and the four-unit factor both match, so this equation defines . Choice D.