How do you tell an exponential function from a linear one in a table?

Check how the output changes over equal steps in x. A linear function has a constant difference: each y is the previous plus the same number. An exponential function has a constant ratio: each y is the previous times the same number. Difference means linear, ratio means exponential. A quadratic instead has a constant second difference.

Why is the base of a growth model 1 plus r, not r?

Each period you keep the whole amount and then add the change. A 3 percent increase makes the amount 103 percent of before, which is 1.03 times as large, so the base is 1 plus 0.03. Using 0.03 alone would shrink the amount to 3 percent of itself. For decay, losing 15 percent leaves 85 percent, so the base is 1 minus 0.15, which is 0.85.

What are the explicit formulas for arithmetic and geometric sequences?

An arithmetic sequence has nth term a sub n equals a sub 1 plus (n minus 1) times d, where d is the common difference. A geometric sequence has nth term a sub 1 times r to the (n minus 1), where r is the common ratio. Both are on the formula sheet. The step count is n minus 1 because the first term needs no steps to reach.

How are sequences related to linear and exponential functions?

An arithmetic sequence is a linear function on the term numbers: the common difference acts like the slope. A geometric sequence is an exponential function on the term numbers: the common ratio acts like the base. So a constant difference matches a line and a constant ratio matches an exponential, the same tests used to classify any function.

Does an exponential function always grow faster than a linear one?

Eventually yes, if it is increasing. An exponential function multiplies by a fixed factor each step, so its increases compound and it overtakes any linear function, however steep, given enough steps. It may start below the line, which is why a graph can show the line ahead at first, but in the long run the increasing exponential wins.

How do you choose between a linear, quadratic, or exponential model?

Decide how the quantity changes. A fixed amount per step (5 dollars per item) is linear. A squared relationship or a situation with a maximum or minimum (area, projectile height) is quadratic. A fixed percent per step (3 percent growth, 15 percent depreciation) is exponential. The words per amount, per square, or per percent point you to the family.

VirginiaMaths

Virginia SOL Algebra I: a complete guide to exponential and comparing functions (A.F)

A deep-dive Virginia SOL Algebra I guide to exponential and comparing functions (A.F, part of the Functions and Statistics block): exponential functions, growth and decay models, arithmetic and geometric sequences, and comparing linear, quadratic, and exponential families.

Generated by Claude Opus 4.815 min readA.F.2Updated 2026-06-02

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

Jump to a section

What this category demands
Exponential functions
Exponential growth and decay
Arithmetic and geometric sequences
Comparing function families
How this category is examined
Check your knowledge

What this category demands

This guide covers exponential and comparing functions, part of the A.F standards in the Functions and Statistics block on the Virginia Algebra I SOL. The skills are recognizing and interpreting exponential functions, modeling growth and decay, using sequence formulas, and comparing the three function families. The unifying idea is how a quantity changes, by addition (linear), by a squared turning relationship (quadratic), or by multiplication (exponential). Each dot-point page has its own practice: exponential functions, exponential growth and decay, arithmetic and geometric sequences, and comparing function families.

Exponential functions

An exponential function $f(x) = ab^x$ has initial value $a$ (output at $x = 0$ ) and base $b$ (the constant multiplier, $b > 0$ , $b \ne 1$ ). The signature in a table is a constant ratio: each output is the previous times the same factor. $b > 1$ grows; $0 < b < 1$ decays. The graph is a curve approaching a horizontal asymptote.

Exponential growth and decay

Use growth $y = a(1 + r)^t$ and decay $y = a(1 - r)^t$ , with $a$ the start, $r$ the percent rate as a decimal, and $t$ the time. Build the base from the percent: $5\%$ growth gives $1.05$ ; $15\%$ decay gives $0.85$ . These formulas are not on the sheet.

Arithmetic and geometric sequences

An arithmetic sequence adds a common difference $d$ : $a_n = a_1 + (n - 1)d$ . A geometric sequence multiplies by a common ratio $r$ : $a_n = a_1 \cdot r^{\,n-1}$ . Both are on the formula sheet, and the step count is $(n - 1)$ because the first term needs no steps. Arithmetic sequences are linear; geometric sequences are exponential.

Comparing function families

Classify by how the output changes over equal $x$ -steps: constant difference is linear, constant second difference is quadratic, constant ratio is exponential. Graphs: line, parabola, asymptotic curve. In context, a fixed amount per step is linear, a percent per step is exponential, and a peak or squared relationship is quadratic.

How this category is examined

Multiple choice. Identify a function family, interpret a growth or decay model, or pick the best model.
Fill-in-the-blank. Write an exponential function, evaluate a growth or decay model, or find a sequence term.
Table items. Classify by differences or ratios, extend a sequence, or match representations.

Check your knowledge

Work these as you would for credit on the online test.

A table shows $3, 6, 12, 24$ . Write the exponential function. (2 points)
In $f(x) = 500(0.8)^x$ , what does $500$ represent? (1 point)
A town of $8000$ grows $3\%$ per year. Write the population function. (2 points)
A car worth $\$ 20{,}000 $loses$ 15%$ each year. Write its value function. (1 point)
Find the $20$ th term of $7, 11, 15, 19, \dots$ . (2 points)
Which is geometric: $3, 6, 12, 24$ or $3, 6, 9, 12$ ? (1 point)
Find the $6$ th term of $5, 10, 20, 40, \dots$ . (2 points)
A table has $y = 4, 12, 36, 108$ . Which family? (1 point)
A base of $0.90$ in a decay model is what percent decrease? (1 point)
A savings account earns $2\%$ per year. Linear or exponential? (1 point)

Solutions

Step 1: Q1. Write an exponential function from a table

The first value gives $a = 3$ and the constant ratio gives $b = 2$ :

f(x) = 3 \cdot 2^x

Step 2: Q2. Interpret the parameter a

In $f(x) = ab^x$ , setting $x = 0$ gives $f(0) = a$ , so $a$ is the initial value, the output at the start.

Step 3: Q3. Write a growth model

The initial population is $8000$ and the annual growth rate is $3\%$ , so the base is $1 + 0.03 = 1.03$ :

P = 8000(1.03)^t

Step 4: Q4. Write a decay model

The initial value is $20000$ and the annual depreciation rate is $15\%$ , so the base is $1 - 0.15 = 0.85$ :

V = 20000(0.85)^t

Step 5: Q5. Use the arithmetic nth-term formula

The common difference is $d = 4$ and $a_1 = 7$ . The $20$ th term needs $19$ steps from the first:

a_{20} = 7 + 19(4) = 83

Step 6: Q6. Identify the geometric sequence

Check the ratios: $\frac{6}{3} = 2$ , $\frac{12}{6} = 2$ , $\frac{24}{12} = 2$ . The constant ratio $2$ confirms that $3, 6, 12, 24$ is geometric; $3, 6, 9, 12$ has a constant difference, so it is arithmetic.

Step 7: Q7. Find a geometric sequence term

The sequence $5, 10, 20, 40, \ldots$ has $a_1 = 5$ and $r = 2$ . The $6$ th term needs the ratio applied $5$ times:

a_6 = 5 \cdot 2^5 = 5 \cdot 32 = 160

Step 8: Q8. Classify by differences and ratios

The ratios $\frac{12}{4} = 3$ , $\frac{36}{12} = 3$ , $\frac{108}{36} = 3$ are constant, so the table represents an exponential function.

Step 9: Q9. Read the percent decrease from the base

A decay base of $0.90$ means the quantity retains $90\%$ each period, so the decrease is:

1 - 0.90 = 0.10, \quad \text{that is } 10\%

Step 10: Q10. Choose the function family

A savings account earns interest as a fixed percent of the current balance each period, so the balance multiplies by a constant factor. That is the signature of an exponential function, not a linear one.

Sources & how we know this

2023 Mathematics Standards of Learning — Virginia Department of Education (2023)
Algebra I Formula Sheet — Virginia Department of Education (2023)