SAT Math · Exponential Functions

Master Exponential
Functions

Learn the concept, study examples, then tackle 20 questions — from easy to expert.

What is an Exponential Function?

An exponential function has the general form:

\( f(x) = a \cdot b^x \)
\(a\) — the coefficient (also called the initial value). It's the value of \(f(0)\), because \(f(0) = a \cdot b^0 = a \cdot 1 = a\).
\(b\) — the base (also called the growth/decay factor). Must be positive and \(b \neq 1\).
If \(b > 1\): exponential growth  |  If \(0 < b < 1\): exponential decay

Finding f(0) — The Key Trick

The problem asks: "which equation shows the value of c as the coefficient or the base?"

Since \(f(0) = c\), substitute \(x = 0\) and simplify. Whatever constant is left equals \(c\).

\( f(0) = a \cdot b^{g(0)} = c \)

The trick is that any base raised to 0 equals 1, so \(b^0 = 1\), leaving only the coefficient.

Worked Examples

Example A
\( f(x) = 22(1.5)^{x+1} \)

\( f(0) = 22(1.5)^{0+1} = 22(1.5)^1 = 33 \)
c = 33 as the product — c is neither the coefficient (22) nor the base (1.5) alone.
Example B
\( f(x) = 33(1/2)^x \)

\( f(0) = 33(1/2)^0 = 33 \cdot 1 = 33 \)
c = 33, which IS the coefficient. ✓
Example C
\( f(x) = 49.5(1.5)^{x-1} \)

\( f(0) = 49.5(1.5)^{-1} = 49.5 \cdot \frac{1}{1.5} = 33 \)
c = 33, but not the coefficient (49.5) or base (1.5).
Example D
\( f(x) = 74.25(1.5)^{x-2} \)

\( f(0) = 74.25 \cdot (1.5)^{-2} = \frac{74.25}{2.25} = 33 \)
c = 33, but not the coefficient or base.

All four give f(0) = 33, but only B has c = 33 appearing as the coefficient itself.

Question 1 of 20 Easy
0
out of 20

Well done!

Here's how you did.

Correct Answers—
Wrong Answers—
Score—
Review 1 of ?