Teacher License – Math 2 Complete Review Guide 2026–2027

4. If \(x+\frac{1}{x}=k\) then \(x^{2}+\frac{1}{x^{2}}=k^{2}-2\) and \(x^{3}+\frac{1}{x^{3}}=k^{3}-3k\). Example: \(k=5\) gives \(23\), then \(x^{6}+\frac{1}{x^{6}}=23^{3}-3\cdot 23=12098\).
5. The roots of \(ax^{2}+bx+c=0\) are \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\); with \(\Delta=b^{2}-4ac\): positive gives two distinct real roots, zero a repeated root, negative no real roots.
6. For a quadratic, the sum of the roots is \(-\frac{b}{a}\) and the product is \(\frac{c}{a}\); the vertex of \(y=ax^{2}+bx+c\) is at \(x=-\frac{b}{2a}\).
7. Absolute value: \(|x|=a\) with \(a>0\) means \(x=\pm a\); \(|x|<a\) means \(-a<x<a\); \(|x|>a\) means \(x<-a\) or \(x>a\).
8. A system of two linear equations is solved by elimination or substitution: one solution if the lines intersect, none if parallel, infinitely many if they coincide. For inequalities, shade the intersection of regions; the boundary is solid for \(\le,\ge\) and dashed for \(<,>\).
9. Arithmetic sequence: \(a_{n}=a_{1}+(n-1)d\) and \(S_{n}=\frac{n}{2}(a_{1}+a_{n})\). Geometric sequence: \(a_{n}=a_{1}r^{n-1}\) and \(S_{n}=\frac{a_{1}(1-r^{n})}{1-r}\); if \(|r|<1\) the infinite sum is \(\frac{a_{1}}{1-r}\).
10. Two matrices can be added or subtracted only if their dimensions match; \(A_{m\times n}\) times \(B_{n\times p}\) is defined and has dimension \(m\times p\). Matrix multiplication is associative and distributive but not commutative.
11. For \(A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\) the determinant is \(ad-bc\) and \(A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\) exists only when the determinant is not zero; the system \(AX=B\) has solution \(X=A^{-1}B\).
12. A function pairs each element of the domain with exactly one element of the range, which the vertical line test shows on a graph. A polynomial has domain \(\mathbb{R}\), \(\sqrt{g(x)}\) needs \(g(x)\ge 0\), and a rational function excludes the zeros of its denominator.
13. The exponential \(a^{x}\) (\(a>0,\ a\neq 1\)) has domain \(\mathbb{R}\) and range \((0,\infty)\); \(\log_{a}x\) has domain \((0,\infty)\) and range \(\mathbb{R}\). The range of \(\sin x\) and \(\cos x\) is \([-1,1]\) with period \(2\pi\), and \(\tan x\) has period \(\pi\).
14. The domain of \(f\circ g\) is the set of \(x\) in the domain of \(g\) where \(g(x)\) lies in the domain of \(f\); \(\frac{f}{g}\) excludes zeros of \(g\). A function has an inverse if it is one-to-one; swap \(x\) and \(y\) and solve, domain and range swap, and the graphs are symmetric about \(y=x\).
15. Logarithm rules: \(\log_{a}(xy)=\log_{a}x+\log_{a}y\), \(\log_{a}\frac{x}{y}=\log_{a}x-\log_{a}y\), \(\log_{a}x^{n}=n\log_{a}x\), \(\log_{a}1=0\), \(\log_{a}a=1\), and change of base \(\log_{a}x=\frac{\ln x}{\ln a}\). Check roots of a log equation because the argument must be positive.
16. The linear function \(y=mx+b\) has slope \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) and crosses the \(y\)-axis at \(b\); parallel lines have equal slopes, and perpendicular lines have slopes whose product is \(-1\). In a word problem define the variables, write the equation, then check the answer.

Common mistakes in this chapter

• ✕ Multiplying or dividing an inequality by a negative number reverses the sign: from \(-2x>6\) we get \(x<-3\), not \(x>-3\).
• ✕ In a rational equation, check the roots after solving: a value that makes any denominator zero is extraneous even if the steps were correct.
• ✕ \(\log_{a}(x+y)\) is not \(\log_{a}x+\log_{a}y\); the correct rule is \(\log_{a}(xy)=\log_{a}x+\log_{a}y\). Likewise \((a+b)^{2}\neq a^{2}+b^{2}\).
• ✕ Matrix multiplication is not commutative: usually \(AB\neq BA\), and \((AB)^{-1}=B^{-1}A^{-1}\) reverses the order.
• ✕ \((f\circ g)(x)\) applies \(g\) first and then \(f\), not the other way round, and usually \(f\circ g\neq g\circ f\). The symbol \(f^{-1}(x)\) does not mean \(\frac{1}{f(x)}\).
• ✕ For \(|x|=a\) with \(a<0\) there is no solution, and \(|x|<a\) with \(a<0\) has none either; many answers write \(x=\pm a\) without checking the sign of \(a\).
• ✕ In exponential or logarithmic inequalities, a base between 0 and 1 reverses the inequality, and the argument of a logarithm must be positive.
• ✕ Counting elements of a union: adding only \(n(A)+n(B)\) counts the intersection twice, so subtract \(n(A\cap B)\). The empty set \(\varnothing\) is not \(\{0\}\).