Louisiana LEAP 2025 Algebra I: a complete guide to functions (F-IF, F-BF, F-LE)

What this module covers

This guide covers functions on the Louisiana LEAP 2025 Algebra I test, the largest part of the Major Content category: function notation, domain, and range (F-IF.A), interpreting key features of graphs (F-IF.B.4), average rate of change (F-IF.B.6), building linear and exponential functions (F-BF.A.1), arithmetic and geometric sequences (F-BF.A.2), and comparing linear, quadratic, and exponential families (F-LE). Each dot-point page has its own practice: function notation, domain, and range, interpreting key features, average rate of change, building functions, arithmetic and geometric sequences, and comparing function families.

Function notation, domain, and range

A function gives exactly one output per input. Function notation $f(x)$ names that output; $f(3)$ means substitute $3$ for $x$. The domain is the set of inputs ($x$-values), the range the set of outputs ($y$-values).

Interpreting key features

Read the $y$-intercept (starting value), the zeros / $x$-intercepts (where output is $0$), increasing and decreasing intervals (rises or falls left to right), and the maximum or minimum (vertex). Interpret each in context with units.

Average rate of change

The average rate of change over $[a, b]$ is $\frac{f(b) - f(a)}{b - a}$, the slope between the endpoints. For a line it is constant; for a curve it depends on the interval.

Building functions

Find the starting value and the kind of change: a constant amount added is linear ($f(x) = b + mx$); a constant factor multiplied is exponential ($f(x) = a \cdot b^x$, growth factor $1 + r$, decay $1 - r$).

Sequences

An arithmetic sequence adds a common difference $d$: $a_n = a_1 + (n - 1)d$. A geometric sequence multiplies by a common ratio $r$: $a_n = a_1 r^{\,n-1}$. Both formulas are on the reference sheet, and both use $n - 1$. Arithmetic sequences are linear; geometric are exponential.

Comparing function families

Tell the families apart by what is constant over equal steps: first difference constant is linear, ratio constant is exponential, second difference constant is quadratic. A key fact: exponential growth eventually exceeds linear growth.

How this module is examined

• Equation response. Evaluate a function, compute a rate of change, or find a sequence term.
• Type II reasoning. Interpret a feature or rate, or explain why exponential beats linear.
• Type III modeling. Build a function and use it to predict.
• Multiple choice and graphing. Classify families; read features and domain/range from graphs.

Check your knowledge

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

1. For $f(x) = 3x^2 - 5$, find $f(-2)$. (2 points)
2. Give the domain of $\{(1, 2), (3, 4), (5, 6)\}$. (1 point)
3. A downward parabola peaks at 36 ft at 1.5 s. What does the maximum represent? (2 points)
4. A graph rises between $x = 1$ and $x = 4$. What is this interval called? (1 point)
5. For $f(x) = x^2$, find the average rate of change from $x = 1$ to $x = 4$. (2 points)
6. A pool has 50 gallons, filled at 8 gal/min. Write $f(t)$. (2 points)
7. A count starts at 100 and doubles each hour. Write $f(h)$. (2 points)
8. Find the 20th term of $4, 7, 10, 13, \ldots$. (2 points)
9. Find the 6th term of $3, 6, 12, 24, \ldots$. (2 points)
10. Equally spaced outputs are $3, 6, 12, 24$. Which family? (2 points)