ACT Math Advanced Concepts Sheet

What You Need to Know

ACT Math Advanced Concepts = the “higher-algebra / precalc-lite” skills that show up most often in the hardest third of the test: functions, polynomials, quadratics, radicals/rational exponents, complex numbers, exponentials & logs, sequences, matrices, and trig modeling.

Why it matters: these questions are usually high-point, time-consuming traps unless you recognize the pattern fast and apply the right rule.

Core idea: most problems reduce to one of these moves:

Rewrite (factor, common denominator, exponent/log rules)
Solve (quadratic, system, exponential/log equation)
Interpret (function notation, domain/range, transformations)
Model (sequence rule, trig ratio/law)

Critical reminder: the ACT loves extraneous solutions from squaring, taking roots, and using logs. Always plug back if you manipulated the equation.

Step-by-Step Breakdown

1) Quadratics & “quadratic-like” equations

Get to standard form: $ax^2+bx+c=0$ (or substitute to make it quadratic-like).
Choose the fastest solve:
- Factor if easy (integer roots).
- Otherwise use quadratic formula.
Check domain restrictions (especially if you squared or had radicals/denominators).

Mini example (quadratic-like): Solve $x^4-5x^2+4=0$ .

Let $u=x^2$ ⇒ $u^2-5u+4=0$ ⇒ $(u-1)(u-4)=0$ .
$u=1$ or $u=4$ ⇒ $x^2=1$ or $x^2=4$ ⇒ $x=\pm 1,\pm 2$ .

2) Simplifying radicals & rational exponents

Rewrite roots as exponents: $\sqrt[n]{a^m}=a^{m/n}$ .
Use exponent rules to combine/simplify.
If asked for a simplified radical, pull out perfect powers.

Mini example: Simplify $\sqrt{50}$ .

$\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}$ .

3) Solving exponential & logarithmic equations

A. Exponential equations

Isolate the exponential expression.
Match bases if possible: $a^{f(x)}=a^{g(x)} \Rightarrow f(x)=g(x)$ (for $a>0$ , $a\neq 1$ ).
If bases don’t match, take logs and solve.

B. Log equations

Use log properties to combine into a single log if helpful.
Convert to exponential form: $\log_a(M)=N \Rightarrow a^N=M$ .
Apply domain: log arguments must be positive: $M>0$ .

Mini example (log): Solve $\log_2(x-1)=3$ .

$2^3=x-1$ ⇒ $8=x-1$ ⇒ $x=9$ . Check: $x-1=8>0$ ok.

4) Function questions (notation, composition, inverse, transformations)

Translate notation: $f(3)$ means plug in $x=3$ .
Composition: $(f\circ g)(x)=f(g(x))$ .
Inverse: swap $x$ and $y$ , solve for $y$ , then rename $y=f^{-1}(x)$ .
Transformations (graph shifts/stretches): compare to $y=f(x)$ .

Mini example (inverse): Find inverse of $f(x)=3x-7$ .

Let $y=3x-7$ . Swap: $x=3y-7$ .
Solve: $x+7=3y$ ⇒ $y=\dfrac{x+7}{3}$ .
$f^{-1}(x)=\dfrac{x+7}{3}$ .

5) Sequences (arithmetic, geometric, recursive)

Decide type:
- Constant difference ⇒ arithmetic.
- Constant ratio ⇒ geometric.
Use the right formula (explicit or recursive).
If asked for a far term, explicit is usually fastest.

Mini example: Arithmetic sequence with $a_1=5$ and $d=3$ . Find $a_{20}$ .

$a_n=a_1+(n-1)d$ ⇒ $a_{20}=5+19\cdot 3=62$ .

6) Matrices (when they show up)

For multiplication $AB$ , check dimensions: if $A$ is $m\times n$ and $B$ is $n\times p$ , product is $m\times p$ .
Multiply row-by-column.
For a $2\times 2$ inverse, use determinant and swap/negate rule.

Mini example: If $A=\begin{pmatrix}1&2\\3&4\end{pmatrix}$ , then $\det(A)=1\cdot 4-2\cdot 3=-2$ .

7) Trig in advanced concept style

For right triangles: use $\sin,\cos,\tan$ definitions.
For non-right triangles: consider Law of Sines / Law of Cosines.
If a question is “height” or “angle of elevation,” set up $\tan(\theta)=\dfrac{\text{opposite}}{\text{adjacent}}$ .

Key Formulas, Rules & Facts

Quadratics & polynomials

Formula / Rule	When to use	Notes
$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$	Solve any quadratic	Discriminant $\Delta=b^2-4ac$ gives # of real roots
Vertex of $y=ax^2+bx+c$ : $x_v=\dfrac{-b}{2a}$	Find max/min or vertex	Then plug in for $y_v$
Vertex form: $y=a(x-h)^2+k$	Graph/transform parabola	Vertex is $(h,k)$
Factoring pattern: $a^2-b^2=(a-b)(a+b)$	Difference of squares	Very common time-saver
If $p(r)=0$ then $(x-r)$ is a factor	Factor theorem	Used with given roots
Remainder theorem: remainder of division by $(x-c)$ is $p(c)$	“Remainder when …”	Instant plug-in
Sum/product of roots for $ax^2+bx+c$	If roots are $r,s$	$r+s=\dfrac{-b}{a}$ , $rs=\dfrac{c}{a}$

Exponents, radicals, rational expressions

Rule	When to use	Notes
$a^m\cdot a^n=a^{m+n}$	Combine like bases	Same base required
$\dfrac{a^m}{a^n}=a^{m-n}$	Simplify quotient	$a\neq 0$
$(a^m)^n=a^{mn}$	Power of a power	Watch parentheses
$a^{-n}=\dfrac{1}{a^n}$	Negative exponents	Move across fraction bar
$a^{m/n}=\sqrt[n]{a^m}$	Convert radicals ↔ exponents	For even $n$ , require $a\ge 0$ in real numbers
Rational expression restriction	Any time you have a denominator	Denominator $\neq 0$ (state excluded values)
Rationalizing	If answer choices are rationalized	Multiply by conjugate: $a+\sqrt{b}$ with $a-\sqrt{b}$

Complex numbers

Fact	When to use	Notes
$i^2=-1$	Simplify powers of $i$	Cycle: $i^1=i$ , $i^2=-1$ , $i^3=-i$ , $i^4=1$
$\sqrt{-a}=i\sqrt{a}$ for $a>0$	Simplify radicals with negatives	Keep $i$ outside
Conjugates: $a+bi$ and $a-bi$	Division / rationalizing complex denom	Product: $(a+bi)(a-bi)=a^2+b^2$

Logs & exponentials

Rule	When to use	Notes
$\log_a(xy)=\log_a(x)+\log_a(y)$	Expand or combine logs	Requires $x>0,y>0$
$\log_a\left(\dfrac{x}{y}\right)=\log_a(x)-\log_a(y)$	Division inside log	Requires $x>0,y>0$
$\log_a(x^k)=k\log_a(x)$	Bring exponent down	Requires $x>0$
$\log_a(a^x)=x$ and $a^{\log_a(x)}=x$	Inverses	$a>0,a\neq 1, x>0$
Change of base: $\log_a(x)=\dfrac{\log_b(x)}{\log_b(a)}$	Calculator has only $\log$ or $\ln$	Choose $b=10$ or $b=e$
Growth/decay: $A=A_0(1+r)^t$	Percent change over time	$r$ as decimal; decay if $r<0$

Functions

Concept	What it means	Notes
Domain	Allowed $x$ -values	Avoid zero denominators; even-root radicands $\ge 0$ ; log arguments $>0$
Range	Possible $y$ -values	Often from graph or algebra
Composition	$(f\circ g)(x)=f(g(x))$	Substitute carefully
Inverse	Undo the function	Not every function has an inverse unless one-to-one (often implied)
Average rate of change	$\dfrac{f(b)-f(a)}{b-a}$	Slope of secant line

Sequences

Type	Key formulas	Notes
Arithmetic	$a_n=a_1+(n-1)d$	$d$ is common difference
Arithmetic sum	$S_n=\dfrac{n}{2}(a_1+a_n)$	Or $S_n=\dfrac{n}{2}(2a_1+(n-1)d)$
Geometric	$a_n=a_1\cdot r^{n-1}$	$r$ is common ratio
Geometric sum	$S_n=a_1\dfrac{1-r^n}{1-r}$	For $r\neq 1$

Matrices (most common ACT facts)

Operation	Rule	Notes
Add/subtract	Same dimensions; add entries	Only defined for same size
Multiply	Row-by-column	Dimensions must match
Determinant $2\times 2$	If $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ , then $ad-bc$	Used for invertibility
Inverse $2\times 2$	$A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$	Only if $ad-bc\neq 0$

Trig essentials (advanced concept style)

Tool	Formula	When to use
Right-triangle trig	$\sin(\theta)=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos(\theta)=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan(\theta)=\dfrac{\text{opp}}{\text{adj}}$	Basic trig ratio problems
Pythagorean identity	$\sin^2(\theta)+\cos^2(\theta)=1$	When given $\sin$ or $\cos$ and need the other
Law of Sines	$\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$	Non-right triangles; AAS/ASA/SSA cases
Law of Cosines	$c^2=a^2+b^2-2ab\cos C$	Non-right triangles; SAS or SSS

Examples & Applications

Example 1: Discriminant / number of real solutions

How many real solutions does $2x^2+3x+5=0$ have?

Compute $\Delta=b^2-4ac=3^2-4(2)(5)=9-40=-31$ .
Since $\Delta<0$ , 0 real solutions (two complex).

ACT angle: often asks “no real solutions / one / two” without solving.

Example 2: Radical equation (extraneous risk)

Solve $\sqrt{x+5}=x-1$ .

Domain: need $x-1\ge 0$ ⇒ $x\ge 1$ .
Square both sides: $x+5=(x-1)^2=x^2-2x+1$ .
$0=x^2-3x-4=(x-4)(x+1)$ ⇒ $x=4$ or $x=-1$ .
Check domain: only $x=4$ works.

ACT angle: one choice is the extraneous root.

Example 3: Log properties / solve

Solve $\log_3(x)+\log_3(x-2)=2$ .

Combine: $\log_3(x(x-2))=2$ .
Convert: $x(x-2)=3^2=9$ ⇒ $x^2-2x-9=0$ .
$x=\dfrac{2\pm\sqrt{4+36}}{2}=1\pm\sqrt{10}$ .
Domain: $x>0$ and $x-2>0$ ⇒ $x>2$ ⇒ only $x=1+\sqrt{10}$ .

Example 4: Function composition & interpretation

Given $f(x)=x^2-1$ and $g(x)=2x+3$ , find $(f\circ g)(2)$ .

First: $g(2)=2\cdot 2+3=7$ .
Then: $f(7)=7^2-1=48$ .

ACT angle: many students mistakenly compute $(g\circ f)(2)$ instead.

Common Mistakes & Traps

Forgetting domain restrictions
- Wrong: solving and keeping values that make a denominator $0$ , a log argument $\le 0$ , or an even root radicand $<0$ .
- Fix: write restrictions early (excluded values), and re-check at the end.
Dropping parentheses in exponent/radical problems
- Wrong: treating $(-3)^2$ as $-3^2$ .
- Fix: remember $-3^2=-(3^2)=-9$ but $(-3)^2=9$ .
Creating extraneous solutions by squaring
- Wrong: solving $\sqrt{\; }$ equations and keeping all algebraic roots.
- Fix: after squaring, plug back into the original equation.
Misusing log rules
- Wrong: $\log(a+b)=\log(a)+\log(b)$ .
- Fix: logs split only for multiplication/division, not addition/subtraction.
Composition/inverse confusion
- Wrong: $(f\circ g)(x)=f(x)\circ g(x)$ or swapping order.
- Fix: do it inside-out: compute $g(x)$ first, then feed into $f$ .
Assuming every function has an inverse on all real numbers
- Wrong: inverting a parabola without restricting domain.
- Fix: if the problem doesn’t state a restriction, it usually gives a one-to-one function (often linear). If it’s not one-to-one, look for “restricted domain” in the prompt.
Sequence indexing errors
- Wrong: using $a_n=a_1+nd$ instead of $a_n=a_1+(n-1)d$ .
- Fix: check with $n=1$ ; your formula must return $a_1$ .
Matrix multiplication order
- Wrong: assuming $AB=BA$ .
- Fix: matrices generally do not commute; follow the order given.

Memory Aids & Quick Tricks

Trick / Mnemonic	Helps you remember	When to use
SOH-CAH-TOA	$\sin=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan=\dfrac{\text{opp}}{\text{adj}}$	Right-triangle trig setups
Discriminant sign	$\Delta>0$ two real, $\Delta=0$ one real (double), $\Delta<0$ none real	“How many real solutions?”
“Swap & solve” for inverse	Swap $x$ and $y$ , solve for $y$	Inverse function problems
i-power cycle (1, i, -1, -i)	Fast simplify $i^n$	Complex numbers
Check with $n=1$	Fix sequence off-by-one	Any explicit sequence formula
Conjugate cleanup	$(a+\sqrt{b})(a-\sqrt{b})=a^2-b$ and $(a+bi)(a-bi)=a^2+b^2$	Rationalizing radicals/complex division

Quick Review Checklist

You can solve quadratics by factoring or $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ and use $\Delta=b^2-4ac$ to count real roots.
You simplify radicals by pulling out perfect squares and convert between $\sqrt[n]{\;}$ and rational exponents $a^{m/n}$ .
You solve exponential equations by matching bases or using logs; you solve log equations by rewriting in exponential form and enforcing $\text{argument}>0$ .
You handle functions: evaluate $f(a)$ , compose $(f\circ g)(x)$ , and find inverses by swapping $x,y$ .
You know arithmetic vs geometric sequences and can use $a_n=a_1+(n-1)d$ or $a_n=a_1r^{n-1}$ .
You can do basic matrix operations and know $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$ .
You can set up trig using SOH-CAH-TOA and use Law of Sines/Cosines when it’s not a right triangle.
You always check for extraneous solutions after squaring or using logs.

One last pass through these rules and you’ll pick up a lot of “hard” points quickly.