Exponential Growth and Decay on the Digital SAT

Quick Answer: Exponential growth and decay models describe quantities that increase or decrease by a constant percentage over time. Use the built-in Desmos calculator on the Digital SAT to quickly graph these functions and identify initial values or growth rates.

graph LR
    A[Read Question] --> B[Identify Initial Value] --> C[Determine Rate] --> D[Build Function] --> E[Solve for Target]

What Is Exponential Growth and Decay?

Exponential functions model situations where a quantity changes by a constant percentage over equal time intervals. Unlike linear equations that change by a constant amount, exponential functions multiply by a constant factor. You will frequently see these functions on the College Board Digital SAT, particularly in the Advanced Math domain.

The standard form of an exponential equation is $y = a(b)^x$. In this formula, $a$ represents the initial value (or y-intercept), and $b$ represents the growth or decay factor. If $b > 1$, the function represents exponential growth. If $0 < b < 1$, it represents exponential decay. While you might be used to solving complex polynomial equations using the /sat/math/quadratic-formula or analyzing parabolas by converting to /sat/math/vertex-form-quadratic, exponential functions require a different approach focused on percentage changes and exponents.

Because these equations can get calculation-heavy, leveraging the Desmos Calculator built into the Bluebook testing app is a highly effective strategy for verifying your models and finding intersections quickly.

Step-by-Step Method

1. Step 1 — Identify the initial amount ($a$). This is the starting value when time $t = 0$.
2. Step 2 — Determine if the quantity is increasing (growth) or decreasing (decay) to choose between adding or subtracting your rate.
3. Step 3 — Convert the given percentage rate to a decimal ($r$). For example, 8% becomes 0.08.
4. Step 4 — Calculate the multiplier factor ($b$). Use $1 + r$ for growth and $1 - r$ for decay.
5. Step 5 — Set up the equation $y = a(b)^t$ and plug in the given time to solve for the final amount.

Desmos Shortcut

Instead of calculating large exponents by hand, type the exponential equation directly into the built-in Desmos calculator. For example, if you need to find when a population reaches 5,000, type your model like y = 200(1.05)^x on line 1, and y = 5000 on line 2. Click the intersection point on the graph to instantly find the exact time ($x$) it takes to reach that target. You can also use the table feature to see the exact value of the function at specific integer intervals.

Worked Example

Question: A population of 500 bacteria grows by 12% every hour. Which function models the population, $P(t)$, after $t$ hours?

A) $P(t) = 500(0.12)^t$ B) $P(t) = 500(1.12)^t$ C) $P(t) = 12(500)^t$ D) $P(t) = 500(0.88)^t$

Solution:

First, identify the initial value. The starting population is 500, so $a = 500$.

Next, identify the rate and convert it to a decimal. The growth rate is 12%, which is: $r = \frac{12}{100} = 0.12$

Since the population is growing, we add this rate to 1 to find the growth factor: $b = 1 + 0.12 = 1.12$

Finally, plug these values into the standard exponential formula $P(t) = a(b)^t$: $P(t) = 500(1.12)^t$

This matches option B.

Correct Answer: B

Common Traps

1. Confusing growth and decay factors — Based on Lumist student data, 60% of students initially confuse the growth factor ($1+r$) with the decay factor ($1-r$). If a problem says a value loses 15% of its value, the multiplier is $1 - 0.15 = 0.85$, not 1.15.

2. Forgetting to convert percentages to decimals — Our analytics show that on compound interest and exponential problems, 25% of errors happen because students forget to convert the percentage to a decimal. Always divide the percentage by 100 before adding or subtracting it from 1.

FAQ

What is the formula for exponential growth and decay?

The standard formula is $y = a(1 \pm r)^t$, where $a$ is the initial amount, $r$ is the rate as a decimal, and $t$ is time. Use the plus sign for growth and the minus sign for decay.

How do I know if an equation represents growth or decay?

Look at the base of the exponent, also known as the multiplier. If the base is greater than 1, it represents exponential growth. If it is between 0 and 1, it represents exponential decay.

How do I convert a percentage to a decimal for these equations?

Divide the percentage by 100 or move the decimal point two places to the left. For example, a 5% growth rate becomes 0.05, making the growth factor 1.05.

How many Exponential Growth and Decay questions are on the SAT?

Advanced Math makes up a significant portion of the Digital SAT Math section, representing some of the more challenging concepts on the test. On Lumist.ai, we have 35 practice questions specifically covering exponential growth and decay to help you master this topic.