Pre-Calculus All-in-One For Dummies Cheat Sheet

Quadratic formula

Quadratic formula applies to a quadratic equation of the form $a{x}^2+bx+c=0$ and gives you the solutions (two, one, or no real solutions) using the following formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Area of a triangle

Area of a triangle can be determined if you have the base and height, the measures of the three sides, or the measure of an angle and the two sides that form that angle:

$A=\frac{1}{2}bh$(b: base, h: height)

$A=\sqrt{s(s-a)(s-b)(s-c)}$ (s: semi-perimeter; a, b, c: sides)

$A=\frac{1}{2}absinC,A=\frac{1}{2}bcsinA,A=\frac{1}{2}acsinB$ (a, b, c: sides; A, B, C: angles)

Sequences and series

A sequence (list of numbers) and series (sum of a list of numbers) can come in many different forms. The most commonly used formulas involve arithmetic sequences (where terms are the same difference apart) and geometric sequences (where each term is a multiple of the previous term).

Arithmetic sequence, where n is the term number, $a_1$ is the first term, and d is the common difference:

$a_n=a_1+(n-1)d$

Sum of k terms of an arithmetic series:

$S_k={\sum }_{n=1}^k{a}_n=\frac{k}{2}(a_1+a_k)$

Geometric sequence, where n is the term number, $g_1$ is the first term, and r is the common ratio:

$g_n=g_1{r}^{n-1}$

Sum of k terms of a series:

$S_k={\sum }_{n=1}^k{g}_1\cdot r^{n-1}=g_1\left(\frac{1-r^k}{1-r}\right)$

Combinations (n things taken r at a time):

$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$

Binomial theorem (used to expand the power of a binomial):

${(a+b)}^n=$${\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k=$$\left(\genfrac{}{}{0ex}{}{n}{0}\right)a^n{b}^0+\left(\genfrac{}{}{0ex}{}{n}{1}\right)a^{n-1}{b}^1+\left(\genfrac{}{}{0ex}{}{n}{2}\right)a^{n-2}{b}^2+\cdots +\left(\genfrac{}{}{0ex}{}{n}{n-2}\right)a^2{b}^{n-2}+\left(\genfrac{}{}{0ex}{}{n}{n-1}\right)a^1{b}^{n-1}+\left(\genfrac{}{}{0ex}{}{n}{n}\right)a^0{b}^n$

Laws of sines and cosines

Laws of sines and cosines apply to the angles A, B, and C of a triangle and the sides a, b, and c opposite those respective angles.

Law of sines:

$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

Laws of cosines:

$a^2=b^2+c^2-2bccosA$

$b^2=a^2+c^2-2accosB$

$c^2=a^2+b^2-2abcosC$

Pre-calculus basic graphing techniques

When working in mathematics, it’s really helpful to draw a picture. In pre-calculus, you have graphs of polynomials, conics, and trig functions to assist you in accomplishing your goals.

Linear and polynomial functions

Linear and polynomial functions are all smooth curves that move from negative infinity to positive infinity.

Linear functions are straight lines with a slope, m, telling you whether it is rising, falling, horizontal, or vertical.

Standard form where $m=-\frac{A}{B}$:

$Ax+By=C$

Slope-intercept form with m (slope) and b (y-intercept):

$y=mx+b$

Quadratic functions are parabolas (U-shaped curves) that open upward or downward.

Polynomial form with vertex $(-\frac{b}{2a},f(-\frac{b}{2a}))$ ; opening upward when $a>0$ and downward when $a<0$:

$y=a{x}^2+bx+c$

Standard form with vertex $(h,k)$; opening upward when $a>0$ and downward when $a<0$:

$y=a{(x-h)}^2+k$

Polynomial functions are smooth curves with a maximum of $n-1$ turning points.

Polynomial form with y-intercept $(0,a_0)$:

$y=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\cdots +a_1{x}^1+a_0$

Conic sections

Conic sections are curves formed when one or two cones are sliced vertically, horizontally, or on a slant.

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

${(x-h)}^2+{(y-k)}^2=r^2$

Parabola with vertex $(h,k)$:

$y-k=a{(x-h)}^2$

Ellipse with center $(h,k)$:

$\frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{b^2}=1$

Hyperbolas with center $(h,k)$:

$\frac{{(x-h)}^2}{a^2}-\frac{{(y-k)}^2}{b^2}=1$ has asymptotes $y=\pm \frac{b}{a}(x-h)+k$

$\frac{{(y-k)}^2}{a^2}-\frac{{(x-h)}^2}{b^2}=1$ has asymptotes $y=\pm \frac{a}{b}(x-h)+k$

Sine and cosine functions

Sine and cosine functions are smooth curves repeating their function values infinitely.

Showing $\left|a\right|$ (amplitude), $\frac{2\pi }{b}$ (period), $-\frac{c}{\left|b\right|}$ (phase shift), d (midline).

Sine: $y=asin(bx+c)+d$

Cosine: $y=acos(bx+c)+d$

Graph transformations

Graph transformations are performed on a function’s basic graph that can make it steeper or flatter, move it up, down, left, or right, and reflect it over a vertical or horizontal line. The original graph keeps its basic properties, making the indicated adjustments. Using the standard format for a function, $y=a\cdot f(x\pm h)\pm k$.

Stretching and flattening makes a graph steeper or flatter:

$a\cdot f\left(x\right)$ stretches when $a>1$, and flattens when $0<a<1$.

Horizontal shifts:

$f(x\pm h)$ moves h units left when $+h$, and moves h units right when $-h$.

Vertical shifts:

$f\left(x\right)\pm k$ moves k units up when $+k$, and moves k units down when $-k$.

Horizontal reflections:

$-f\left(x\right)$ reflects over a horizontal line.

Vertical reflections:

$f(-x)$reflects over a vertical line.

Trig identities in pre-calculus

Trigonometric identities are equations or statements that are true for the angle measures inserted in the variables. Identities are helpful when you want to simplify a complex equation or solve for some particular function value. Here are the most commonly used trig identities found in pre-calculus and calculus.

Pythagorean identities

Pythagorean identities refer to a right triangle and its angles:

${sin}^{2}x+{cos}^{2}x=1$

${tan}^{2}x+1={sec}^{2}x$

$1+{cot}^{2}x={csc}^{2}x$

Reciprocal identities

Reciprocal identities describe the algebraic relationship between reciprocal functions:

$cscx=\frac{1}{sinx}$

$secx=\frac{1}{cosx}$

$cotx=\frac{1}{tanx}$

Double-angle identities

Double-angle identities allow you to find the trig function values of angles that correspond to angles twice the size of known angle values:

$sin2x=2\cdot sinx\cdot cosx$

$cos2x={cos}^{2}x-{sin}^{2}x=1-2{sin}^{2}x=2{cos}^{2}x-1$

$tan2x=\frac{2tanx}{1-{tan}^{2}x}$

Half-angle identities

Half-angle identities give you the function values of angles that are half the size of known angle values: $sin\left(\frac{x}{2}\right)=\pm \sqrt{\frac{1-cosx}{2}}$

$cos\left(\frac{x}{2}\right)=\pm \sqrt{\frac{1+cosx}{2}}$

$tan\left(\frac{x}{2}\right)=\frac{1-cosx}{sinx}$

Even-odd identities

Even-odd identities determine if a function is even or odd and apply that particular property:

$sin(-x)=-sinx$

$cos(-x)=cosx$

$tan(-x)=-tanx$

Sum and difference identities

Sum and difference identities take known function values and use them to find the value of an angle that is the sum or difference of known angle values:

$sin(x\pm y)=sinxcosy\pm cosxsiny$

$cos(x\pm y)=cosxcosy\mp sinxsiny$

$tan(x\pm y)=\frac{tanx\pm tany}{1\mp tanxtany}$

Product-to-sum identities

Product-to-sum identities change expressions that are products to sums and differences of new function measures:

$sinx\cdot siny=\frac{1}{2}[cos(x-y)-cos(x+y)]$

$cosx\cdot cosy=\frac{1}{2}[cos(x-y)+cos(x+y)]$

$sinx\cdot cosy=\frac{1}{2}[sin(x+y)+sin(x-y)]$

$cosx\cdot siny=\frac{1}{2}[sin(x+y)-sin(x-y)]$

Sum-to-product identities

Sum-to-product identities take sums or differences of function values and change them to products of new function values:

$sinx\pm siny=2sin\left(\frac{x\pm y}{2}\right)\cdot cos\left(\frac{x\mp y}{2}\right)$

$cosx+cosy=2cos\left(\frac{x+y}{2}\right)\cdot cos\left(\frac{x-y}{2}\right)$

$cosx-cosy=-2sin\left(\frac{x+y}{2}\right)\cdot sin\left(\frac{x-y}{2}\right)$

Co-function identities

Co-function identities describe the relationship between functions of angles in a right triangle:

$sin(\frac{\pi }{2}-x)=cosx$

$cos(\frac{\pi }{2}-x)=sinx$

$tan(\frac{\pi }{2}-x)=cotx$

$cot(\frac{\pi }{2}-x)=tanx$

$sec(\frac{\pi }{2}-x)=cscx$

$csc(\frac{\pi }{2}-x)=secx$