Algebra 2
10 questions · Functions, Logs, Sequences & more
Vertex & Completing the Square
If \(f(x) = x^2 - 6x + 5\), what is the minimum value of \(f(x)\)?
KEY:vertex x = −b/2a → plug back in. Or complete the square: \((x-h)^2 + k\), the min is k.
Worked Example — \(g(x) = x^2 - 4x + 7\)
①vertex: \(x = \frac{-(-4)}{2(1)} = 2\)
②\(g(2) = 4 - 8 + 7 = 3\)
③minimum value = 3
⚠️ Tricky: Students often answer the x-coordinate (3) instead of the value f(3).
Log Laws — Change of Base
Solve: \(\log_2 x = 1 + \log_2 3\)
KEY:log A + log B = log AB — combine logs on the same side, then drop the log.
Worked Example — \(\log_3 x = 1 + \log_3 2\)
①\(\log_3 x - \log_3 2 = 1\)
②\(\log_3 \frac{x}{2} = 1\)
③\(\frac{x}{2} = 3 \Rightarrow x = 6\)
⚠️ Tricky: \(1 = \log_2 2\), so combine: \(\log_2 x = \log_2(2 \cdot 3)\).
Arithmetic Sequence — nth Term
The 3rd term of an arithmetic sequence is 11 and the 7th term is 27. Find the common difference.
KEY:Between term 3 and term 7 = 4 gaps. difference = \(\frac{a_7 - a_3}{7-3}\)
Worked Example — 2nd term = 5, 6th term = 21
①gaps = 6 − 2 = 4
②\(d = \frac{21-5}{4} = 4\)
⚠️ Count the gaps, not terms. From term 3 to term 7 is 4 gaps, not 5.
Simplifying Rational Expressions
Simplify: \(\dfrac{x^2 - 9}{x^2 - x - 6}\)
KEY:Factor both top and bottom, then cancel common factors. \(a^2-b^2 = (a+b)(a-b)\)
Worked Example — \(\frac{x^2-4}{x^2-x-2}\)
①Top: \((x+2)(x-2)\)
②Bottom: \((x-2)(x+1)\)
③Cancel: \(\dfrac{x+2}{x+1}\)
The simplified form is \(\dfrac{x+3}{x+2}\). What is the numerator's constant?
Solving Exponential Equations — Same Base
Solve: \(2^{x-1} = 16\)
KEY:Make same base → set exponents equal. \(16 = 2^?\)
Worked Example — \(3^{x+1} = 27\)
①\(27 = 3^3\)
②\(x + 1 = 3 \Rightarrow x = 2\)
⚠️ Tricky: \(2^{x-1} = 2^4\) means \(x - 1 = 4\), so \(x = 5\). Don't forget to add 1!
Composition of Functions
If \(f(x) = 2x + 1\) and \(g(x) = x - 3\), find \(f(g(4))\).
KEY:inside out — evaluate \(g(4)\) first, then plug result into \(f\).
Worked Example — \(f(x)=3x, g(x)=x+2\), find \(f(g(1))\)
①\(g(1) = 1+2 = 3\)
②\(f(3) = 3 \times 3 = 9\)
⚠️ Don't do \(g(f(4))\) — order matters!
Remainder Theorem
What is the remainder when \(p(x) = x^3 - 2x^2 + x - 3\) is divided by \((x - 2)\)?
KEY:Dividing by \((x-a)\)? Just compute p(a). That's the remainder. No long division needed!
Worked Example — \(q(x)=x^2+3x-1\) ÷ \((x-1)\)
①\(q(1) = 1 + 3 - 1 = 3\)
②remainder = 3
⚠️ Dividing by \((x-2)\) means plug in \(x = +2\), not \(-2\).
Sum of Infinite Geometric Series
Find the sum: \(8 + 4 + 2 + 1 + \cdots\)
KEY:Infinite geo sum = a ÷ (1−r), only works when \(|r| < 1\). Here \(r = \frac{1}{2}\).
Worked Example — \(6 + 2 + \frac{2}{3} + \cdots\)
①\(a = 6,\ r = \frac{1}{3}\)
②\(S = \frac{6}{1 - \frac{1}{3}} = \frac{6}{\frac{2}{3}} = 9\)
⚠️ Formula only applies when \(-1 < r < 1\). Always check r first!
Multiplying Complex Numbers
Simplify: \((1 + i)^2\)
KEY:Expand like normal algebra, then replace i² = −1. That's the only special rule.
Worked Example — \((2+i)(2-i)\)
①\(= 4 - 2i + 2i - i^2\)
②\(= 4 - (-1) = 5\)
The real part of \((1+i)^2\) is 0. What is the imaginary coefficient?
Matrix Determinant — 2×2
Find \(\det\begin{pmatrix}4 & 2\\1 & 1\end{pmatrix}\)
KEY:2×2 det = ad − bc. Multiply diagonal down-right, subtract diagonal down-left.
Worked Example — \(\det\begin{pmatrix}3 & 1\\2 & 5\end{pmatrix}\)
①\(ad = 3 \times 5 = 15\)
②\(bc = 1 \times 2 = 2\)
③\(15 - 2 = 13\)
⚠️ Order matters: it's \(ad - bc\), NOT \(bc - ad\).