What are the exponential growth and decay models?

Growth is y equals a times the quantity 1 plus r, all to the power t. Decay is y equals a times the quantity 1 minus r, to the power t. Here a is the initial value at time zero, r is the rate as a decimal, and the quantity in parentheses is the growth or decay factor. These are not on the reference sheet, so memorize them, and never put the bare rate in the base: 5 percent growth uses base 1.05, not 0.05.

What is the growth or decay factor?

It is the base of the exponential, the number you multiply by each period. For growth it is 1 plus r, which is greater than 1; for decay it is 1 minus r, which is between 0 and 1. A factor of 0.88 means keep 88 percent each period, the same as losing 12 percent. A base above 1 grows, and a base between 0 and 1 decays.

What are the key features of an exponential graph?

For f of x equals a times b to the x, the y-intercept is the point (0, a) because b to the zero is 1. The horizontal asymptote is y equals 0, since the curve approaches but never touches the x-axis. The domain is all real numbers and the range is y greater than 0. A base above 1 grows; a base between 0 and 1 decays.

How do you tell linear, quadratic, and exponential apart from a table?

Check how the outputs change for equal input steps. A constant first difference (adding the same amount) is linear. A constant second difference (the differences of the differences are equal) is quadratic. A constant ratio (multiplying by the same factor) is exponential. Add for linear, accelerate for quadratic, multiply for exponential.

Why does exponential growth eventually beat linear or quadratic?

A linear function adds a fixed amount each step, so its increases stay the same size. An exponential function multiplies by a fixed factor, so each increase is a percentage of an ever-larger amount and the increases themselves grow. The compounding accelerates without bound, so even if the linear plan leads at first, the exponential one overtakes it and stays ahead.

Why is a square root's domain restricted but a cube root's is not?

A real number squared is never negative, so no real number is the square root of a negative; the square-root function only accepts inputs that keep the inside greater than or equal to zero. A cube preserves sign, since negative times negative times negative is negative, so every real number is the cube of some real number, and the cube-root function accepts all inputs. Even roots restrict the domain; odd roots do not.

FloridaMaths

B.E.S.T. Algebra 1 EOC: a complete guide to exponential and nonlinear functions

A deep-dive B.E.S.T. Algebra 1 EOC guide to exponential and nonlinear functions: growth and decay models (MA.912.AR.5), graphing exponentials and their asymptotes, distinguishing linear, quadratic, and exponential families, and square-root, cube-root, and piecewise functions. The models are not on the reference sheet, so memorize them.

Generated by Claude Opus 4.815 min readMA.912.AR.5, MA.912.F.1Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What this strand demands
Exponential growth and decay
Graphing exponentials
Comparing function families
Square-root, cube-root, and piecewise functions
How this strand is examined
Check your knowledge

What this strand demands

This guide covers exponential and nonlinear functions (MA.912.AR.5 and parts of MA.912.F.1), within the Functions and Modeling category. Exponentials power compound interest and depreciation problems, and the function-family comparison is a recurring EOC theme. The other nonlinear shapes, square-root, cube-root, absolute-value, and piecewise, round out the families a student must recognize. Each dot-point page has its own practice: exponential growth and decay, graphing exponential functions, comparing linear, quadratic, and exponential, and square-root, cube-root, and piecewise functions.

Exponential growth and decay

Growth is $y = a(1 + r)^t$ and decay is $y = a(1 - r)^t$ , where $a$ is the initial value, $r$ is the rate (decimal), and the parenthesis is the factor. A factor above $1$ grows; between $0$ and $1$ decays. These are not on the reference sheet, so memorize them, and use the factor ( $1.05$ ), never the bare rate ( $0.05$ ), as the base.

Graphing exponentials

For $f(x) = a \cdot b^x$ : the $y$ -intercept is $(0, a)$ , the horizontal asymptote is $y = 0$ , the domain is all reals, and the range is $y > 0$ . Base $> 1$ rises (growth); base between $0$ and $1$ falls (decay). The curve hugs the asymptote on one end and runs away on the other.

Comparing function families

Identify the family by its pattern of change: constant first difference is linear, constant second difference is quadratic, constant ratio is exponential. Over the long run, a quantity that multiplies (exponential) eventually overtakes one that adds (linear) or grows by squares (quadratic).

Square-root, cube-root, and piecewise functions

A square-root function has a restricted domain (inside $\ge 0$ ) and a half-curve shape. A cube-root function accepts all reals and has an S-shape. An absolute-value function is a V with a corner. A piecewise function uses different rules on different intervals, so evaluate by choosing the piece whose condition the input meets.

How this strand is examined

Equation editor and number entry. Write a growth or decay model, evaluate it, or evaluate a piecewise function.
Multiple choice and editing task. Choose the correct model, identify a $y$ -intercept or asymptote, classify a function family, or state a domain.
GRID and matching. Match equations to curves; plot intercepts or starting points.
Context items. Compare growing quantities and reason about long-run behavior.

Check your knowledge

Work these as you would for credit on the computer-based test.

Write a model for $1500 growing 6 percent per year. (1 point)
A $24,000 truck loses 15 percent of its value yearly. Write the model. (1 point)
What is the $y$ -intercept of $f(x) = 5 \cdot 3^x$ ? (1 point)
Does $f(x) = 8 \cdot (0.5)^x$ grow or decay, and what is its horizontal asymptote? (2 points)
Outputs $2, 10, 50, 250$ for inputs $0, 1, 2, 3$ . Which family? (1 point)
Outputs $3, 7, 11, 15$ . Which family? (1 point)
State the domain of $f(x) = \sqrt{x - 9}$ . (1 point)
For $f(x) = x - 1$ if $x < 2$ and $f(x) = 4x$ if $x \ge 2$ , find $f(0)$ and $f(5)$ . (2 points)

Solutions

Step 1: Write an exponential growth model for an investment

The growth formula is $y = a(1 + r)^t$ , where $a$ is the initial amount, $r$ is the annual rate as a decimal, and $t$ is time in years. Use the factor $1 + r$ as the base, not the bare rate $r$ .

V = 1500(1.06)^t

Final answer: $V = 1500(1.06)^t$ .

Step 2: Write an exponential decay model for a depreciating truck

For depreciation, value is lost each year, so use the decay formula $y = a(1 - r)^t$ . Losing 15 percent per year means keeping 85 percent, which gives a factor of $1 - 0.15 = 0.85$ .

V = 24000(0.85)^t

Final answer: $V = 24000(0.85)^t$ .

Step 3: Find the y-intercept of an exponential function

The $y$ -intercept is the output when $x = 0$ . Any base raised to the power zero equals 1, so $f(0) = a \cdot b^0 = a$ . The coefficient $a$ is always the $y$ -intercept for functions of the form $a \cdot b^x$ .

f(0) = 5 \cdot 3^0 = 5 \cdot 1 = 5 \implies (0, 5)

Final answer: $(0, 5)$ .

Step 4: Classify growth or decay and state the asymptote

A base between 0 and 1 produces decay because each multiplication shrinks the output. The horizontal asymptote is $y = 0$ because the function approaches zero but never reaches it.

Base $0.5 < 1$ , so $f(x) = 8 \cdot (0.5)^x$ is a decay function; horizontal asymptote $y = 0$ .

Final answer: Decay; horizontal asymptote $y = 0$ .

Step 5: Identify the function family from a table of outputs

To classify a table, check what operation produces each successive output. If dividing consecutive outputs gives the same number each time, the pattern is a constant ratio and the function is exponential.

Ratios: $10 \div 2 = 5$ , $50 \div 10 = 5$ , $250 \div 50 = 5$ . Constant ratio of 5.

Final answer: Exponential.

Step 6: Identify the function family from a table with a constant difference

A constant first difference means each output increases by the same amount, which is the defining property of a linear function.

Differences: $7 - 3 = 4$ , $11 - 7 = 4$ , $15 - 11 = 4$ . Constant first difference of 4.

Final answer: Linear.

Step 7: State the domain of a square-root function

The expression inside a square root must be greater than or equal to zero because the square root of a negative number is not a real number. Set the inside expression at least equal to zero and solve for $x$ .

x - 9 \ge 0 \implies x \ge 9

Final answer: $x \ge 9$ .

Step 8: Evaluate a piecewise function at two inputs

To evaluate a piecewise function, first determine which piece's condition the input satisfies, then apply only that rule. Using the wrong piece is the most common error.

$f(0)$ : $0 < 2$ , so use $f(x) = x - 1$ : $f(0) = 0 - 1 = -1$ .

$f(5)$ : $5 \ge 2$ , so use $f(x) = 4x$ : $f(5) = 4(5) = 20$ .

Final answer: $f(0) = -1$ ; $f(5) = 20$ .

Sources & how we know this

B.E.S.T. Mathematics Standards — Florida Department of Education (2020)
B.E.S.T. Algebra 1 EOC Computer-Based Practice Test — Florida Department of Education (2024)