Ohio Algebra I: a complete guide to functions

What this category demands

This guide covers functions, the largest reporting category on Ohio's Algebra I test, drawn from F-IF (interpreting functions), F-BF (building functions), and F-LE (linear, quadratic, and exponential models). It rewards reading, building, and comparing functions, not just computing. Each dot-point page has its own practice: function notation, domain, and range, interpreting key features, average rate of change, building and writing functions, arithmetic and geometric sequences, exponential functions, growth, and decay, and comparing function families.

Function notation, domain, and range

A function assigns each input one output; the vertical line test confirms this on a graph. The notation $f(x)$ is the output at input $x$: evaluate $f(4)$ by substituting, solve $f(x) = k$ for the input. The domain is the inputs, the range the outputs, and a context can restrict the domain to, say, whole numbers.

Key features and rate of change

Read a graph by its intercepts, increasing and decreasing intervals, maximum or minimum, and positive and negative regions, all described in terms of the input $x$. The average rate of change over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$, the slope of the secant line, constant for a line, varying for a curve.

Building functions and sequences

To build a linear function $f(x) = mx + b$, read the rate as the slope and the starting value as the intercept, or find the slope from two points then the intercept. Sequences are functions of the term number: arithmetic adds a common difference ($a_n = a_1 + (n-1)d$, like linear), geometric multiplies by a common ratio ($a_n = a_1 r^{\,n-1}$, like exponential). Both formulas are on the reference sheet.

Exponentials and comparing families

An exponential function is $f(x) = ab^x$: $a$ the initial value, $b$ the base ($b > 1$ grows, $0 < b < 1$ decays). The percentage models, growth $y = a(1 + r)^t$ and decay $y = a(1 - r)^t$, are not on the reference sheet. To compare families from a table: constant first difference is linear, constant second difference is quadratic, constant ratio is exponential, and for large inputs the order is exponential > quadratic > linear.

How this category is examined

• Numeric and equation response. Evaluate or solve $f(x)$, compute a rate or a sequence term, or write a function rule.
• Multiple choice and multiple-select. Decide whether a relation is a function, read a key feature, or classify a family.
• Tables and graphs. Complete a function table, read intercepts or intervals, or compute differences and ratios.

Check your knowledge

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

1. If $f(x) = 2x - 7$, find $f(5)$ and solve $f(x) = 1$. (2 points)
2. A graph crosses the $x$-axis at $-2$ and $6$. State the zeros. (1 point)
3. $g(1) = 4$ and $g(5) = 28$. Find the average rate of change over $[1, 5]$. (2 points)
4. A printer prints $20$ pages already and adds $8$ per minute. Write $P(t)$ and find $P(5)$. (2 points)
5. Find the $7$th term of the arithmetic sequence $3, 7, 11, \ldots$. (2 points)
6. Write a growth model for \400$increasing$5%$ per year. (2 points)
7. A table's outputs are $6, 18, 54, 162$. Which family? (1 point)