Write and interpret exponential functions for growth and decay, identify the initial value and growth factor, and contrast exponential change (constant ratio) with linear change (constant difference).

What this topic is asking

The Functions category requires you to model with exponential functions (the F-LE and F-IF standards). On the Grade 10 MCAS you write a function for growth or decay, read its initial value and growth factor, and contrast exponential behavior with linear. The distinction (constant ratio versus constant difference) is a favorite MCAS comparison, so understanding why exponentials behave differently matters as much as the formula.

The form of an exponential function

An exponential function is built around repeated multiplication:

• $A_0$ is the initial value, the output when $x = 0$ (since $b^0 = 1$). In a context it is the starting amount.
• $b$ is the base, the factor the quantity is multiplied by each time $x$ increases by 1.

So $f(x) = 500(1.08)^x$ starts at 500 and multiplies by 1.08 every step.

Growth versus decay

The base $b$ decides the behavior:

• Growth when $b > 1$: the quantity increases. The base is $1 + r$, where $r$ is the growth rate. A base of $1.08$ is 8% growth.
• Decay when $0 < b < 1$: the quantity decreases. The base is $1 - r$, where $r$ is the decay rate. A base of $0.88$ is 12% decay.

The reliable move is to compare the base with 1 and then find $r$ as the distance from 1. A base of $0.95$ is a 5% decay; a base of $1.20$ is a 20% growth. Reading the base itself as the percentage is the classic error.

Exponential versus linear

The MCAS frequently asks you to tell exponential and linear apart from a table or a description. The test is in the differences and ratios between successive outputs:

• Linear: equal steps in $x$ give a constant difference in $y$ (you add the same amount). Slope is that common difference per unit.
• Exponential: equal steps in $x$ give a constant ratio in $y$ (you multiply by the same factor).

For the table $x = 0, 1, 2, 3$ with $y = 3, 6, 12, 24$: the differences are $3, 6, 12$ (not constant) but the ratios are all $2$ (constant), so the function is exponential with base 2. If instead $y = 3, 7, 11, 15$, the differences are all $4$, so it is linear. A key consequence the MCAS highlights: an exponential eventually outgrows any linear function, no matter how steep the line.

Compound interest as an exponential model

A common MCAS context is compound interest, which is exponential growth in disguise. An amount $P$ growing at annual rate $r$ for $t$ years becomes $A = P(1 + r)^t$, exactly the growth form with base $1 + r$. For \$1000 at 4$A = 1000(1.04)^3 = 1000(1.124864) \approx \$1124.86$. The reference sheet provides simple interest, but compound interest follows this exponential pattern, so recognizing it as $P(1 + r)^t$ is the key.

The same structure handles half-life style decay. If a quantity halves each period, the base is $\frac{1}{2}$: $A = A_0\left(\frac{1}{2}\right)^t$. After 3 periods, only $\left(\frac{1}{2}\right)^3 = \frac{1}{8}$ remains. Whether the context is money, population, or medicine, the same form $A_0 b^t$ applies, with the base built from the percent change.

Reading an exponential graph

An exponential graph either rises steeply (growth) or falls toward, but never reaches, a horizontal line (decay). The graph crosses the y-axis at the initial value $A_0$, because $b^0 = 1$. A growth curve gets steeper as it goes right; a decay curve flattens as it approaches zero. The MCAS may show such a graph and ask for the initial value (the y-intercept) or whether the function grows or decays (the direction), both read directly from the curve.

Try this

Q1. A savings account of $1000 grows 3% per year. Write the value function.

• Cue. $V = 1000(1.03)^t$.

Q2. A table shows $y$-values $5, 15, 45, 135$ for $x = 0, 1, 2, 3$. Linear or exponential?

• Cue. Constant ratio of 3, so exponential.