Course Review

Equations: isolate the variable by undoing operations in reverse order --- distribute, clear fractions with the LCD, gather variable terms on one side, divide by the coefficient. Keep every step equivalent.

Literal equations: same moves, but treat the other letters as constants. If the target variable appears in two terms, factor it out first: $ax+bx=c \Rightarrow x=\dfrac{c}{a+b}$.

Inequalities: solve like equations, but flip the sign when you multiply or divide by a negative.

Absolute value: $|x|$ is distance from $0$ (never negative). Isolate $|\cdot|$ first, then split.

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• $|ax+b|=c\ (c>0)$: $ax+b=c$ or $ax+b=-c$.
• $|ax+b|<c$: “less thAND” $-c<ax+b<c$.
• $|ax+b|>c$: “greatOR” $ax+b<-c$ or $ax+b>c$.

multicols2

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• $x<a$: $(-\infty,\,a)$
• $x\le a$: $(-\infty,\,a]$
• $x>a$: $(a,\,\infty)$
• $x\ge a$: $[a,\,\infty)$
• AND: $a<x<b \to (a,b)$
• OR: $(-\infty,a)\cup(b,\infty)$

multicols Use a parenthesis next to $\pm\infty$ --- infinity is never included.

A relation pairs inputs with outputs; a function assigns exactly one output to each input (passes the vertical line test). Domain = allowed inputs $x$; range = resulting outputs $y$.

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• Exclude $x$ that make a denominator zero.
• Exclude $x$ that make an even radicand negative (need radicand $\ge 0$).

multicols2

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• Linear: $y=x$
• Quadratic: $y=x^2$
• Cubic: $y=x^3$
• Absolute value: $y=|x|$
• Square root: $y=\sqrt{x}$
• Reciprocal: $y=\tfrac{1}{x}$
• Exponential: $y=b^x$
• Logarithmic: $y=\log_b x$

multicols

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• $h$: horizontal shift (right if $h>0$);   $k$: vertical shift (up if $k>0$).
• $|a|>1$ vertical stretch, $0<|a|<1$ vertical shrink; $a<0$ reflects over the $x$-axis.
• $b$ affects horizontal scaling by factor $\tfrac{1}{|b|}$; $b<0$ reflects over the $y$-axis.

Composition: $(f\circ g)(x)=f\big(g(x)\big)$ --- do $g$ first (inside out).

Inverse $f^{-1}$: swap $x$ and $y$, then solve for $y$. Graphs reflect over $y=x$; a one-to-one function (horizontal line test) has an inverse. Check: $f\big(f^{-1}(x)\big)=x$.

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• Substitution: solve one equation for a variable, substitute.
• Elimination: add multiples of equations to cancel a variable.
• 3-variable: eliminate one variable twice to reduce to a $2\times2$ system, then back-substitute.

Outcomes: one solution (lines cross), no solution (parallel, $0=k$), infinitely many (same line, $0=0$).

Add/subtract entrywise (same dimensions). Scalar $c$: multiply every entry.

Multiplication $AB$: needs (columns of $A$) $=$ (rows of $B$); entry $=$ row $\cdot$ column dot product. Not commutative.

$2\times2$: $\det\begin{bmatrix} a & b\\ c & d \end{bmatrix}=ad-bc$.   Inverse: $A^{-1}=\dfrac{1}{ad-bc}\begin{bmatrix} d & -b\\ -c & a \end{bmatrix}$ (exists iff $\det\ne 0$).

Cramer's Rule for $\begin{cases} ax+by=e\\ cx+dy=f \end{cases}$ with $D=ad-bc\ne0$:

Replace the variable's column with the constants.

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• Standard: $y=ax^2+bx+c$; vertex at $x=-\dfrac{b}{2a}$; $y$-intercept $(0,c)$.
• Vertex: $y=a(x-h)^2+k$; vertex $(h,k)$; axis $x=h$.
• Factored: $y=a(x-r_1)(x-r_2)$; zeros $r_1,r_2$.

Opens up if $a>0$, down if $a<0$.

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• Factoring / Zero Product
• Square roots
• Completing the square
• Quadratic formula

multicols

Discriminant $D=b^2-4ac$: $D>0$ two real roots; $D=0$ one repeated real root; $D<0$ two complex conjugate roots.

$i=\sqrt{-1}$, $i^2=-1$. Powers cycle with period 4:

Add/subtract: combine real and imaginary parts. Multiply: FOIL, replace $i^2=-1$. Divide: multiply by the conjugate $a-bi$.

For degree $n$ with leading coefficient $a$:

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• $n$ even, $a>0$: both ends up.   $n$ even, $a<0$: both ends down.
• $n$ odd, $a>0$: down-left, up-right.   $n$ odd, $a<0$: up-left, down-right.

A degree-$n$ polynomial has at most $n$ real zeros and $n-1$ turning points.

Synthetic division divides by $(x-c)$: bring down, multiply by $c$, add.

Remainder Theorem: the remainder of $P(x)\div(x-c)$ equals $P(c)$.

Factor Theorem: $(x-c)$ is a factor $\iff P(c)=0$.

Rational Root Theorem: any rational zero $=\dfrac{p}{q}$, where $p\mid$ (constant term), $q\mid$ (leading coefficient).

Fundamental Theorem of Algebra: a degree-$n$ polynomial has exactly $n$ complex roots (counting multiplicity). Complex and irrational roots come in conjugate pairs.

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• Multiply: factor, cancel common factors, multiply across.
• Divide: multiply by the reciprocal.
• Add/subtract: get a common denominator (LCD), combine numerators.
• Complex fractions: multiply top and bottom by the overall LCD.

Solving: multiply every term by the LCD to clear fractions, then solve. Check for extraneous roots (any value making a denominator $0$ is rejected).

For $f(x)=\dfrac{p(x)}{q(x)}$ in lowest terms:

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• Vertical asymptote: where $q(x)=0$ (and $p\ne0$ there).
• Hole: at a factor that cancels from top and bottom.
• Horizontal asymptote: compare degrees --- deg top $<$ bottom: $y=0$; equal: $y=\dfrac{\text{lead}}{\text{lead}}$; top $>$ bottom by 1: slant asymptote (by division).

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• Direct: $y=kx$.   Inverse: $y=\dfrac{k}{x}$.
• Joint: $y=kxz$.   Combined: $y=\dfrac{kx}{z}$.

Find $k$ from a known data point, then answer the question.

$\sqrt[n]{a}=a^{1/n}$,   $a^{m/n}=\sqrt[n]{a^m}=\big(\sqrt[n]{a}\big)^m$.

Even index: radicand must be $\ge 0$; principal root is nonnegative. Odd index: any real radicand.

Exponent rules still apply: multicols2

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• $a^m a^n=a^{m+n}$
• $\dfrac{a^m}{a^n}=a^{m-n}$
• $(a^m)^n=a^{mn}$
• $a^{-n}=\dfrac{1}{a^n}$

multicols

Solve: isolate the radical, raise both sides to the index power, solve, then check for extraneous roots (squaring can introduce false solutions).

The inverse of $y=x^2$ ($x\ge0$) is $y=\sqrt{x}$; more generally squaring and square-rooting undo each other on the appropriate domain.

$y=b^x$ ($b>0,\ b\ne1$): growth if $b>1$, decay if $0<b<1$; horizontal asymptote $y=0$. The number $e\approx 2.71828$.

Logarithm is the inverse: $\log_b y=x \iff b^x=y$.   $\ln x=\log_e x$.   $\log x=\log_{10}x$.

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• Product: $\log_b(MN)=\log_b M+\log_b N$
• Quotient: $\log_b\!\dfrac{M}{N}=\log_b M-\log_b N$
• Power: $\log_b(M^p)=p\log_b M$
• Change of base: $\log_b x=\dfrac{\ln x}{\ln b}$

multicols Solving: exponential equations --- take a log of both sides (or equate bases). Log equations --- rewrite in exponential form, then check the domain (arguments must be $>0$).

multicols2

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• General: $A=A_0(1\pm r)^t$
• Continuous: $A=A_0 e^{rt}$
• Compound: $A=P\Big(1+\dfrac{r}{n}\Big)^{nt}$
• Continuous interest: $A=Pe^{rt}$

multicols Here $r$ = rate (decimal), $n$ = compounds per year, $t$ = years.

Add a constant $d$ each term.

Infinite geometric sum converges only when $|r|<1$:

Multiply by a constant $r$ each term.

$k$ is the index, the bottom value is the start, the top value is the end. Useful facts:

Circle: center $(h,k)$, radius $r$:

Parabola (vertex $(h,k)$, $p$ = vertex-to-focus distance):

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• Opens up/down: $(x-h)^2=4p(y-k)$; focus $(h,k+p)$, directrix $y=k-p$.
• Opens left/right: $(y-k)^2=4p(x-h)$; focus $(h+p,k)$, directrix $x=h-p$.

Ellipse ($a>b$, $a$ = semi-major): $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$, foci on major axis with $c^2=a^2-b^2$.

Hyperbola: $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$ (opens left/right), with $c^2=a^2+b^2$ and asymptotes $y-k=\pm\dfrac{b}{a}(x-h)$.

Eccentricity $e=\dfrac{c}{a}$: circle $e=0$; ellipse $0<e<1$; parabola $e=1$; hyperbola $e>1$.

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• Only one squared term $\to$ parabola.
• $A=C$ (same sign) $\to$ circle.
• $A\ne C$, same sign $\to$ ellipse.
• Opposite signs $\to$ hyperbola.

Fundamental Counting Principle: multiply the choices at each stage.

$P(E)=\dfrac{\text{favorable}}{\text{total}}$, with $0\le P(E)\le1$; complement $P(E')=1-P(E)$.

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• Addition: $P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$ (subtract $0$ if mutually exclusive).
• Multiplication (independent): $P(A\text{ and }B)=P(A)\cdot P(B)$.
• Conditional: $P(B\mid A)=\dfrac{P(A\text{ and }B)}{P(A)}$.

For $n$ independent trials, success probability $p$, exactly $k$ successes:

Empirical (68--95--99.7) Rule for a normal distribution: about $68\%$ of data lie within $1\sigma$ of the mean, $95\%$ within $2\sigma$, $99.7\%$ within $3\sigma$.

$z$-score: $z=\dfrac{x-\bar{x}}{\sigma}$ (number of standard deviations from the mean).