SAT Exponential Functions on Desmos: Solve From a Table

To solve an SAT exponential table question, find how the output changes from one row to the next. If f(1) = a⁵, f(2) = a⁹ and f(3) = a¹³, the exponent rises by 4 each step, so f(x) = a^{4x + 1}. Then f(k) = a²⁹ means 4k + 1 = 29, so k = 7.

You can type each Desmos line on this page into the College Board version of Desmos on our Desmos on the SAT page. This page works one SAT-style problem by hand and on Desmos, walks you through a second, and gives you two to try on your own.

The rule for exponential tables

An exponential function multiplies its output by the same number each time x goes up by 1. You can write it as f(x) = c(r)^x, where c is f(0) and r is that growth factor.

When every output is a power of the same base a, multiplying by a⁴ adds 4 to the exponent. So the exponents follow a straight-line pattern, and you can solve for k with a one-step linear equation.

One more fact finishes the job. If a is greater than 1, then a^m = aⁿ only when m = n. That is why these questions say a > 1. If a were 1, every power of a would equal 1 and any k would work.

Worked example: find f(x), then k (I do)

For the exponential function f, the table shows three values of x and their corresponding values of f(x), where a is a constant greater than 1. If k is a constant and f(k) = a²⁹, what is the value of k?

x	f(x)
1	a5
2	a9
3	a13

SAT-style question written for this page.

Step 1: confirm the growth factor. Divide each output by the one before it. a⁹ / a⁵ = a⁴ and a¹³ / a⁹ = a⁴. The factor is the same both times, so f is exponential with growth factor a⁴.

Step 2: find f(0). The table starts at x = 1, not x = 0. Step back once by dividing: f(0) = a⁵ / a⁴ = a. Now f(x) = a(a⁴)^x = a^{4x + 1}. Check the last row: 4(3) + 1 = 13, which matches a¹³.

Step 3: match exponents. f(k) = a^{4k + 1} = a²⁹. Because a > 1, the exponents must be equal: 4k + 1 = 29, so 4k = 28 and k = 7.

Check it on Desmos, two ways

A Desmos regression needs numbers, and this table has a letter in it. You can work around that in two ways.

Way 1: fit the exponents. The exponents 5, 9 and 13 follow a line, so fit a line to them. Line 1 is a table:

2  y_1 ~ mx_1 + b
3  (29 - b)/m

Line 2 fits m = 4 and b = 1, so the exponent is 4x + 1. Line 3 uses those stored values and returns 7.

Way 2: pick a value for a. The question only says a > 1, so k can't depend on which a you choose. Try a = 2. The outputs become 2⁵ = 32, 2⁹ = 512 and 2¹³ = 8,192. Start a blank graph and make this table on line 1:

2  y_1 ~ cr^{x_1}
3  f(x) = cr^x
4  f(7)
5  2^29

Line 2 fits c = 2 and r = 16. That matches the hand work: c = a = 2 and r = a⁴ = 16. Lines 4 and 5 both give 536,870,912, so f(7) = 2²⁹ and k = 7 holds. The tilde on line 2 takes the place of the equals sign, as Desmos's help article on regressions describes. In the College Board version, Desmos uses Log mode for exponential regressions. With exact data like this table, the fit matches the hand values.

Use c and r for the fitted numbers, not a. Desmos stores each fitted value in the letter you typed, and this question already uses a for something else.

Guided example: numbers instead of letters (we do)

For the exponential function g, g(1) = 12, g(2) = 36 and g(3) = 108. If g(k) = 2,916, what is the value of k?

SAT-style question written for this page.

Step 1 (shown). 36 / 12 = 3 and 108 / 36 = 3, so the growth factor is 3. One step back from x = 1 gives g(0) = 12 / 3 = 4. So g(x) = 4(3)^x.

Step 2 (shown). On Desmos, put 1, 2, 3 in x₁ and 12, 36, 108 in y₁. Then type y_1 ~ cr^{x_1}. Desmos fits c = 4 and r = 3, the same numbers you found by hand.

Step 3 (you finish). Set 4(3)^k = 2,916 and solve for k. Divide by 4 first, then ask which power of 3 you have.

Check your answer: 2,916 / 4 = 729, and 729 = 3⁶, so k = 6. On Desmos, define g(x) = cr^x on a new line and type g(6). It returns 2916.

Your turn: two SAT-style problems (you do)

We wrote these for this page. Solve each one by hand first, then check it on Desmos. The answers are at the end of this section.

Practice 1. For the exponential function h, h(1) = m², h(2) = m⁷ and h(3) = m¹², where m is a constant greater than 1. If h(k) = m⁴⁷, what is the value of k?

Practice 2. For the exponential function p, p(0) = 800, p(1) = 200 and p(2) = 50. For what value of k is p(k) = 3.125?

Answers

1. k = 10. The exponents 2, 7 and 12 rise by 5, so the exponent is 5x - 3. Check x = 1: 5(1) - 3 = 2. Then 5k - 3 = 47, so 5k = 50 and k = 10. On Desmos, y_1 ~ mx_1 + b on the exponents fits m = 5 and b = -3.
2. k = 4. Each step keeps one quarter of the last value, so p(x) = 800(0.25)^x. Then 800 / 3.125 = 256, and 256 = 4⁴, so k = 4. On Desmos, y_1 ~ cr^{x_1} fits c = 800 and r = 0.25.

For linear, quadratic and exponential fits side by side, work through our SAT regression practice problems on Desmos.

When algebra is faster

College Board's calculator policy says some Math questions are better solved without a calculator. We put a table of powers of a in that group. Dividing two outputs and matching exponents takes three short lines, while the Desmos check means typing a table first. Use Desmos when the outputs are messy decimals, or to confirm an answer you already have.

Mistakes to avoid

• Treating the first row as the start. The table begins at x = 1, so f(0) is a, not a⁵. Writing f(x) = a^{4x + 5} gives 4k + 5 = 29 and k = 6, one short of the answer. Check your formula against a second row before you solve.
• Checking with a = 1. Every power of 1 is 1, so the table would read 1, 1, 1 and any k would fit. Pick 2 or 3 for a numeric check.
• Dropping the constant in the exponent. Setting 4k = 29 gives 7.25. When the answer to a "what is k" question comes out as an odd decimal, recheck the exponent formula.

Practice next

Study growth factors in the free Growth Factors and Linear vs Exponential Models lesson notes. For mixed SAT-style math sets, open the TTA SAT question bank.

Exponent rules come up again with roots: see rational exponents and radicals on Desmos. For the parabola version of this skill, read SAT quadratics on Desmos, and for the calculator itself, our College Board Desmos guide.

Checked against Desmos's Regressions help article, Desmos's College Board calculator sheet and College Board's calculator policy on September 26, 2026. Correction: an earlier version of this page gave k = 8 for this table. The table supports k = 7. The Test Advantage is independent of College Board and Desmos.