An exponential function multiplies its output by the same factor each time increases by . Read the marked points as a small data table, then fit an model in Desmos to find its starting value and multiplier. Use the rule for multiplying powers to match an equivalent form. Matching one point alone isn't enough: another curve could grow or shrink at a different rate.
Hints
- Hint 1
In , the initial value is the output at , because . Which marked point gives you that starting value?
- Hint 2
The multiplier tells you how the output changes when increases by . Put the marked points in a Desmos table and fit . What value does Desmos find for the multiplier?
- Hint 3
The model you find may look different from the choices. To move a factor outside the power into the exponent, write that factor as a power of the same base. What happens to the exponents when you multiply?
Step-by-step
Fit the marked points, then rewrite
Step 1Find the starting value
Enter the three marked points into Desmos's table. The row says . For an exponential model , any positive base to the power is :
So the starting value is .
- Step 2
Find the multiplier
Type below the table. The tells Desmos to fit the marked points; under PARAMETERS, it reports . This multiplier means each one-unit increase in triples the output, so .
- Step 3
Write both factors with the same base
Write the starting as so both factors are powers of :
- Step 4
Match the equation to a choice
When you multiply powers with the same base, add their exponents. Add and :
That equation gives the function shown by the graph. Choice A.