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 |
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
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
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 | |
| “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.