Skills

Algebra you need: the one-page sheet

Functions, Graphs and Notation

A function gives each input one output, written f(input) = output . The domain is every allowed input (no zero denominators, no square roots of negatives); the range is every output. Zeros are inputs with output 0 ; the y -intercept is (0, f(0)) . Piecewise-defined functions use the piece whose condition the input meets.

Function: each input gives exactly one output. $f(3) = 7$ means the input $3$ gives the output $7$ .
Domain: the inputs a function accepts. Range: the outputs it produces.
Zero: an input with output $0$ , an $x$ -intercept of the graph. $y$ -intercept: the point $(0, f(0))$ .
To evaluate, replace every $x$ with the input in parentheses: $f(-2)$ , $f(x + 1)$ .

f(3) = 7 names one point (3, 7) on the graph of f the domain leaves out inputs that make a denominator 0 or a square root's inside negative a is a zero of f and the graph meets the x -axis at (a, 0) first find the piece whose condition holds for the input, then use only that piece

function: A rule that gives each input exactly one output. $f(3) = 7$ means the input $3$ gives the output $7$ .
domain: The set of inputs a function accepts. For a formula, every real number that gives a real output: no zero denominators and no square roots of negative numbers.
range of a function: The set of outputs a function produces as the input runs through its domain.
graph of a function: All the points $(x, f(x))$ for $x$ in the domain, drawn in the plane: inputs across, outputs up.
zero of a function: An input $a$ with $f(a) = 0$ . On the graph, a zero is an $x$ -intercept, the point $(a, 0)$ .
y-intercept: The point where a graph crosses the $y$ -axis, $(0, f(0))$ . A function has at most one.
piecewise-defined function: A function that uses different formulas on different parts of its domain. To evaluate it, find the piece whose condition the input meets.

Interval Notation and Inequalities

Interval notation writes a stretch of the number line with its two ends: brackets for included ends, parentheses for excluded ends and for ±∞ , and ∪ to join pieces. Solve linear inequalities like equations, flipping the sign whenever you multiply or divide by a negative, and check one value.

$[a, b]$ includes both ends; $(a, b)$ includes neither; $[a, b)$ includes $a$ but not $b$ .
$\infty$ is not a number you reach, so it always gets a parenthesis: $[2, \infty)$ .
Solve a linear inequality like an equation, but flip the sign when you multiply or divide by a negative.
For a compound inequality $a \le 2x + 3 < b$ , do each step to all three parts.

it gets a bracket, [ or ] , and a closed dot on a graph it gets a parenthesis; ∞ and −∞ always take parentheses the inequality sign flips, because negation reverses order ( 2 < 3 but −2 > −3 ) it is two intervals joined by a union: (−∞, a)∪(a, ∞)

interval notation: A way to write a stretch of the number line by its ends: a bracket for an included end, a parenthesis for an excluded end or for $\pm\infty$ . $[-1, 4)$ means $-1 \le x < 4$ . A closed interval includes both ends; an open interval includes neither.
linear inequality: A statement that one quantity is less than (or greater than) another, such as $3 - 2x > 7$ . A linear inequality, with the variable to the first power, is solved like an equation, but the sign flips when you multiply or divide both sides by a negative number.

Lines, Slope and Linear Equations

Slope is the change in y divided by the change in x , the same between any two points of a line. Point-slope form y − y_1 = m(x − x_1) builds a line from one point and the slope, and y = y_1 + m(x − x_1) gives its value at any x . In context, slope is output units per input unit.

Slope $m$ $= \frac{\text{change in } y}{\text{change in } x}$ : positive rises, negative falls, $0$ is horizontal, a vertical line has none.
Point-slope form $y - y_1 = m(x - x_1)$ ; slope-intercept form $y = mx + b$ with $b$ the $y$ -intercept.
In context the slope has units of output per input, like dollars per minute.

its slope is (y_2 − y_1)/(x_2 − x_1) , rise over run, subtracted in the same order point-slope form y − y_1 = m(x − x_1) gives the line, and y = y_1 + m(x − x_1) gives any value on it y changes by m⋅Δx , the same for every equal step: a line has a constant rate of change

slope: The change in $y$ divided by the change in $x$ between two points of a line, $\frac{y_2 - y_1}{x_2 - x_1}$ . A line has the same slope between any two of its points.
point-slope form: Point-slope form $y - y_1 = m(x - x_1)$ is the line through $(x_1, y_1)$ with slope $m$ ; slope-intercept form $y = mx + b$ has slope $m$ and $y$ -intercept $(0, b)$ .

Quadratics: Zeros and the Quadratic Formula

A quadratic f(x) = ax² + bx + c graphs as a parabola with its vertex at x = −b/2a . Solve f(x) = 0 by factors, by square roots or by the quadratic formula; the discriminant b² − 4ac counts the real zeros. Completing the square gives vertex form a(x − h)² + k , which shows the vertex (h, k) .

A quadratic function is $f(x) = ax^2 + bx + c$ with $a \ne 0$ ; its graph is a parabola.
Solve by factors, by square roots ( $(x - 1)^2 = 4$ gives $x - 1 = \pm 2$ ) or by the quadratic formula.
The vertex is at $x = -\frac{b}{2a}$ ; completing the square gives vertex form $a(x - h)^2 + k$ .

each factor equal to 0 gives a solution the quadratic formula x = (−b ± √(b² − 4ac))/2a gives the solutions there are two, one or no real zeros the vertex is (h, k) , the axis is x = h , and a > 0 opens up, a < 0 opens down

quadratic function: A function $f(x) = ax^2 + bx + c$ with $a \ne 0$ . Its graph is a parabola.
parabola: The U-shaped graph of a quadratic function; it opens up when $a > 0$ and down when $a < 0$ , and is symmetric about the vertical line through its vertex.
vertex of a parabola: The turning point of a parabola, at $x = -\frac{b}{2a}$ . In vertex form $y = a(x - h)^2 + k$ the vertex is $(h, k)$ .
quadratic formula: The solutions of $ax^2 + bx + c = 0$ are $x$ $= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The discriminant $b^2 - 4ac$ tells how many are real: two, one or none. Completing the square rewrites a quadratic in vertex form.

Polynomial Arithmetic and Factoring

Multiply polynomials by distributing every term and combining like terms. Factor by taking out a common factor first, then using a difference of squares a² − b² = (a − b)(a + b) , a sum or difference of cubes, or grouping. A product equals 0 when a factor does, and p(a) = 0 means x − a is a factor.

A polynomial is a sum of terms $ax^n$ with whole-number powers, like $x^3 + x^2 - 7x + 2$ .
Multiply by distributing every term of one factor over every term of the other, then combine like terms.
Factor in order: common factor, then patterns (difference of squares, sum or difference of cubes), then grouping.

take it out first: x³ − 9x = x(x² − 9) use the pattern: x² − 9 = (x − 3)(x + 3) factor by grouping: x²(x + 2) − 4(x + 2) = (x² − 4)(x + 2) each factor equal to 0 gives a solution, and p(a) = 0 means x − a is a factor

polynomial: A sum of terms $ax^n$ with whole-number powers $n$ , such as $x^3 + x^2 - 7x + 2$ .
factoring: Writing an expression as a product: take out a common factor, use patterns such as the difference of squares $a^2 - b^2 = (a - b)(a + b)$ and the sum or difference of cubes, or factor by grouping.

Complex Numbers

The imaginary unit i has i² = −1 , so √(−b²) = bi . Complex numbers a + bi add and multiply like binomials, with i² replaced by −1 . A quadratic with real coefficients and a negative discriminant has two complex solutions that are complex conjugates, a ± bi .

The imaginary unit $i$ has $i^2 = -1$ . A complex number is $a + bi$ with real $a$ (real part) and $b$ (imaginary part).
The complex conjugate of $a + bi$ is $a - bi$ , and $(a + bi)(a - bi)$ $= a^2 + b^2$ , a real number.
Powers of $i$ repeat every four: $i, -1, -i, 1$ .

square roots of negatives exist: √(−16) = 4i work like binomials, then replace i² with −1 its two solutions are complex conjugates a + bi and a − bi

complex number: A number $a + bi$ , where $a$ and $b$ are real and $i$ is the imaginary unit, with $i^2 = -1$ . $a$ is the real part and $b$ the imaginary part.
complex conjugate: The complex conjugate of $a + bi$ is $a - bi$ . Their product, $a^2 + b^2$ , is real.

Rational Expressions

A rational expression is a quotient of polynomials, undefined where its denominator is 0 . Factor, then cancel common factors only, keeping the original exclusions. Add with a common denominator, multiply across, divide by the reciprocal, and split a sum in the numerator term by term.

A rational expression is a quotient of two polynomials, like $\frac{x - 3}{x - 2}$ . It is undefined where its denominator is $0$ .
Factor first, then cancel common factors, never terms. Keep the excluded inputs of the original.
Add with a common denominator; multiply straight across; divide by multiplying by the reciprocal.

a is excluded from the domain, before and after any simplifying it cancels, but the excluded input stays excluded rewrite both over a common denominator, then combine the numerators each term can be divided by the denominator separately: (2x² + 3)/x = 2x + 3/x

rational expression: A quotient of two polynomials, $\frac{p(x)}{q(x)}$ , undefined where $q(x) = 0$ . Simplify by cancelling common factors; the inputs excluded from the original stay excluded.

Systems of Equations

Solve a system by substitution or elimination, removing one variable at a time. Fitting a model through data points means writing one equation per point and solving for the constants, stored at full precision.

A solution of a system makes every equation true at once; for two lines it is where they cross.
To fit $y = ax^2 + bx + c$ to three points, substitute each point to get three equations in $a$ , $b$ and $c$ .
With a calculator, store each constant; rounding early spoils every value that comes after.

a system with as many equations as unknowns one equation in one variable, which you solve all the unknowns, at full precision

system of equations: Two or more equations with the same unknowns, to be solved together. A solution makes every equation true at once. Two lines that cross have one solution; parallel lines have none; the same line has infinitely many.

Exponent and Radical Rules

Same base: add exponents to multiply, subtract to divide, multiply exponents for a power of a power. b^(−n) is a reciprocal and b^(m/n) is a root then a power. Writing both sides with one base turns 4^x = 32 into 2x = 5 .

$b^m \cdot b^n = b^{m + n}$ , $\frac{b^m}{b^n} = b^{m - n}$ , $(b^m)^n = b^{mn}$ , $(ab)^n = a^n b^n$ , $b^0 = 1$ .
$b^{-n} = \frac{1}{b^n}$ : a negative exponent is a reciprocal, never a negative number.
$b^{m/n} = (\sqrt[n]{b})^m$ : the denominator is the root, the numerator the power. Take the root first to keep numbers small.

add the exponents: b^mb^n = b^(m + n) multiply the exponents: (b^m)^n = b^mn take a reciprocal ( b^(−n) = 1/(b^n) ) or a root ( b^(1/n) = n√b )

exponent rules: The laws of exponents for a positive base $b$ : $b^m \cdot b^n = b^{m + n}$ , $\frac{b^m}{b^n} = b^{m - n}$ , $(b^m)^n = b^{mn}$ , $b^0 = 1$ , a negative exponent $b^{-n} = \frac{1}{b^n}$ , and a rational exponent $b^{m/n} = (\sqrt[n]{b})^m$ .
radical: A root written with the radical sign, such as the square root $\sqrt{b}$ or the cube root $\sqrt[3]{b}$ . The $n$ th root of $b$ is the number whose $n$ th power is $b$ , and it equals the rational exponent $b^{1/n}$ .

Right-Triangle Trigonometry

In a right triangle, name the legs from the angle θ : opposite and adjacent. Then sin θ = opposite/hypotenuse , cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent . The ratios depend only on the angle, so one angle and one side are enough to find the other sides.

The hypotenuse is the longest side, so $\sin\theta$ and $\cos\theta$ are between $0$ and $1$ for an acute angle.
$\tan\theta$ $= \frac{\text{opposite}}{\text{adjacent}}$ is rise over run: the slope of the hypotenuse when the adjacent side lies flat.
To find a side: write the ratio, put in the known values, then multiply (unknown on top) or divide (unknown on the bottom).

the leg across from θ is the opposite side, the leg touching θ is the adjacent side, and the side across from the right angle is the hypotenuse the ratios of their sides are the same, whatever the triangle's size, so each ratio depends only on θ sin θ = opposite/hypotenuse , cos θ = adjacent/hypotenuse , tan θ = opposite/adjacent (SOH CAH TOA) pick the ratio that links the known side to the unknown one, write the equation, and solve it

trigonometric ratio: A ratio of two sides of a right triangle, named from an acute angle $\theta$ : $\sin\theta$ $= \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\theta$ $= \frac{\text{adjacent}}{\text{hypotenuse}}$ , $\tan\theta$ $= \frac{\text{opposite}}{\text{adjacent}}$ (SOH CAH TOA). It depends only on the angle, not on the size of the triangle.
hypotenuse: The side of a right triangle across from the right angle, and its longest side. The other two sides, the legs, are named from an acute angle $\theta$ : the opposite side is across from $\theta$ , the adjacent side touches $\theta$ .

Circles, Arcs and Special Triangles

The 45-45-90 triangle has sides 1:1:√2 and the 30-60-90 triangle 1:√3:2 ; they give exact sines and cosines at 30^(∘) , 45^(∘) and 60^(∘) . An arc cut off by a central angle of θ degrees is the fraction θ/360 of the circumference 2πr . The circle with center (h, k) and radius r is (x − h)² + (y − k)² = r² .

45-45-90: sides $1 : 1 : \sqrt{2}$ . 30-60-90: sides $1 : \sqrt{3} : 2$ , with the short leg across from $30^\circ$ .
So $\sin 30^\circ = \frac{1}{2}$ , $\cos 30^\circ = \frac{\sqrt{3}}{2}$ , $\sin 45^\circ$ $= \cos 45^\circ$ $= \frac{\sqrt{2}}{2}$ , $\sin 60^\circ = \frac{\sqrt{3}}{2}$ , $\cos 60^\circ = \frac{1}{2}$ .
Arc length is a fraction of the circumference: $s = \frac{\theta}{360} \cdot 2\pi r$ for an angle of $\theta$ degrees.
The circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$ . The right side is $r^2$ , not $r$ .

with legs 1 , the hypotenuse is √(1² + 1²) = √2 you get a 30-60-90 triangle with short leg 1 , hypotenuse 2 and long leg √(2² − 1²) = √3 its arc has length s = θ/360 ⋅2πr the Pythagorean theorem gives its equation, (x − h)² + (y − k)² = r²

special right triangle: One of the two right triangles whose sides are known exactly from one side: the 45-45-90 triangle, with sides in the ratio $1 : 1 : \sqrt{2}$ , and the 30-60-90 triangle, with sides $1 : \sqrt{3} : 2$ (the short leg across from $30^\circ$ ).
arc length: The length of a piece of a circle. A central angle of $\theta$ degrees cuts off the fraction $\frac{\theta}{360}$ of the circumference, so the arc has length $s = \frac{\theta}{360} \cdot 2\pi r$ .
equation of a circle: The equation $(x - h)^2 + (y - k)^2 = r^2$ of the circle with center $(h, k)$ and radius $r$ : it says the point $(x, y)$ is at distance $r$ from the center. The circle of radius 1 centered at the origin is $x^2 + y^2 = 1$ .