Georgia Milestones Algebra: a complete guide to linear and exponential functions

What this part of the course demands

This guide covers Linear and Exponential Functions, the Functional and Graphical Reasoning (A.FGR) domain, which is about 30 percent of the EOC. It builds on the linear-equation work and pairs each "additive" idea (linear functions, arithmetic sequences) with its "multiplicative" counterpart (exponential functions, geometric sequences), then compares them. Each dot-point page carries its own worked Milestones-style questions: functions, notation, domain, and range, linear functions and rate of change, arithmetic sequences, exponential functions, growth, and decay, geometric sequences, and comparing linear and exponential models.

Functions, notation, domain, and range

A function assigns each input exactly one output; the vertical line test checks this on a graph. Function notation $f(x)$ is the output at input $x$ (substitute, do not multiply). The domain is the inputs, the range is the outputs, and a real context usually restricts both (time $\ge 0$, counts whole).

Linear functions and rate of change

A function is linear exactly when it has a constant rate of change: equal input steps give equal output steps. That rate is the slope $\frac{y_2 - y_1}{x_2 - x_1}$, and for a linear function the average rate of change over any interval equals the slope. Interpret the slope as a per-unit rate and the y-intercept as the starting value.

Arithmetic sequences

An arithmetic sequence adds a common difference $d$. The explicit rule is $a_n = a_1 + (n - 1)d$; the recursive rule is $a_n = a_{n-1} + d$ with $a_1$ stated. An arithmetic sequence is a linear function on the integers, with $d$ as the slope.

Exponential functions, growth, and decay

An exponential function $y = ab^x$ multiplies by base $b$ each step: $b > 1$ grows, $0 < b < 1$ decays. From a percent rate, growth is $y = a(1 + r)^t$ and decay is $y = a(1 - r)^t$. The initial value is $a$; the rate is the base's distance from 1.

Geometric sequences

A geometric sequence multiplies by a common ratio $r$. The explicit rule is $a_n = a_1 \cdot r^{\,n-1}$; the recursive rule is $a_n = r \cdot a_{n-1}$ with $a_1$ stated. A geometric sequence is an exponential function on the integers, with $r$ as the base.

Comparing linear and exponential

Classify a table by constant difference (linear) versus constant ratio (exponential). In a context, a fixed amount per period is linear, a percent per period is exponential. Over the long run, exponential growth always overtakes linear growth, even when it starts behind.

How this strand is examined

• Multiple choice. Evaluate a function, find a rate of change or ratio, classify linear versus exponential, or pick an explicit rule.
• Numeric entry. Find a sequence term or evaluate a growth or decay model.
• Constructed response. Write and interpret a model, or contrast a linear and an exponential situation.

Check your knowledge

Work these as you would for credit on the EOC.

1. If $f(x) = 2x + 3$, find $f(5)$. (1 point)
2. Is $\{(1, 4), (2, 4), (3, 9)\}$ a function? (1 point)
3. A table has $f(0) = 10, f(1) = 7, f(2) = 4$. Find the rate of change. (1 point)
4. Write the explicit rule for the arithmetic sequence $4, 9, 14, 19, \dots$. (1 point)
5. A $5000 investment grows 8 percent per year. Write the model and find the value after 2 years. (2 points)
6. Write the explicit rule for the geometric sequence $2, 8, 32, \dots$. (1 point)
7. A table shows $3, 9, 27, 81$. Linear or exponential, and the key parameter? (1 point)