Equations and Functions on the ACT

How ACT Math Differs from SAT Math

The ACT Math section has 45 questions in 50 minutes — roughly 1 minute per question. Key differences from the SAT:

• 4 answer choices (not 5): A/B/C/D for odd questions, F/G/H/J for even questions
• No grid-in: Every question is multiple choice
• Calculator allowed on all questions (but not all questions need one)
• Broader topic coverage: Includes logarithms, matrices, complex numbers, and Law of Sines/Cosines — none of which appear on the SAT
• Steeper difficulty curve: Questions get progressively harder from Q1 to Q45

Linear Equations

The ACT tests linear equations in two main ways:

1. Solving in context: Set up an equation from a word problem, then solve.
2. Interpreting slope and intercept: Given a model like $y = mx + b$, explain what $m$ and $b$ represent.

Quadratic Equations

You need three solving methods:

Method	When to Use
Factoring	When the expression factors cleanly (e.g., $x^2 - 5x + 6 = 0$)
Quadratic formula	When factoring is messy or coefficients are large
Completing the square	When asked for vertex form or when the problem involves circles

Discriminant shortcut: $b^2 - 4ac$ tells you the number of real solutions without solving:

• Positive → 2 real solutions
• Zero → 1 real solution (repeated root)
• Negative → no real solutions (complex roots — and the ACT does test complex numbers)

Functions

The ACT expects you to:

• Evaluate: Given $f(x) = 3x^2 - 1$, find $f(-2)$. Answer: $3(4) - 1 = 11$.
• Compose: Given $f(x) = x + 1$ and $g(x) = x^2$, find $f(g(3))$. Answer: $f(9) = 10$.
• Interpret domain/range: Which input values are valid? What outputs are possible?
• Recognize transformations: $f(x - 3)$ shifts right 3, $f(x) + 2$ shifts up 2, $-f(x)$ reflects over the x-axis.

Logarithms (ACT-Specific — Not on SAT)

The ACT includes 1–2 logarithm questions. Key rules:

Rule	Example
Definition	$\log_b(a) = c$ means $b^c = a$
Product	$\log(ab) = \log(a) + \log(b)$
Quotient	$\log(a/b) = \log(a) - \log(b)$
Power	$\log(a^n) = n \cdot \log(a)$
Change of base	$\log_b(a) = \frac{\ln(a)}{\ln(b)}$

Sequences and Series (ACT-Specific)

Arithmetic sequence: Common difference $d$. Term: $a_n = a_1 + (n-1)d$. Sum: $S_n = \frac{n}{2}(a_1 + a_n)$.

Geometric sequence: Common ratio $r$. Term: $a_n = a_1 \cdot r^{n-1}$. Sum: $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$.