Exponential Functions — Core Skills

Exponential Functions: The Basics

Part 1 of 2 — One Skill, One Idea

Linear growth adds the same amount each step: $10, 20, 30, 40$. You are adding $10$ every time.

Exponential growth multiplies by the same number each step: $10, 20, 40, 80$. You are doubling every time.

Every exponential function on the SAT looks like this:

$y = a \cdot b^{x}$

• $a$ is the starting amount — the value when $x = 0$.
• $b$ is the base, also called the growth factor — the number you multiply by each step.
• $x$ is the exponent — how many times you multiply.

If $b$ is bigger than $1$, the amount grows. If $b$ is between $0$ and $1$, the amount shrinks. Shrinking is called decay.

The one move: turn a percent into the number $b$

• Growing by $7\%$? Then $b = 1 + 0.07 = 1.07$. You keep all of it and add a little.
• Shrinking by $7\%$? Then $b = 1 - 0.07 = 0.93$. You keep the rest.

Worked example

You put \200$in an account that grows$10%$per year. Write the model and find the amount after$1$ year.

Step 1 — The starting amount is $a = 200$.

Step 2 — Growing $10\%$ means $b = 1 + 0.10 = 1.10$.

Step 3 — Put the pieces in place: $y = 200(1.10)^{t}$, where $t$ is the number of years.

Step 4 — For one year, put $t = 1$ in: $y = 200(1.10)^{1} = 200 \times 1.10 = 220$.

After one year there is \220$ in the account.

Exponential Functions: Practice

Part 2 of 2 — Run the Steps

To build or read an exponential model $y = a \cdot b^{x}$:

1. Find the starting amount. That number goes in front, as $a$.
2. Find the percent change in the problem.
3. Turn the percent into the base $b$. Growing: $b = 1 + \text{percent as a decimal}$. Shrinking: $b = 1 - \text{percent as a decimal}$.
4. Put the number of steps in for the exponent and do the arithmetic.

Reading a model at a glance

• $b > 1$ means growth. Example: $1.09$ is growth of $9\%$.
• $0 < b < 1$ means decay. Example: $0.88$ is a drop of $12\%$.
• The $y$-intercept is always the point $(0, a)$, because the exponent is $0$ there and $b^{0} = 1$.

Percent as a decimal

Move the decimal point two places left.

• $5\% = 0.05$
• $25\% = 0.25$
• $40\% = 0.40$

One year of growth is one multiplication. Two years is two multiplications, and so on.