SAT Facts and Formulas

Numbers

• Even numbers: $\ldots, -6, -4, -2, 0, 2, 4, 6, \ldots , 2n, \ldots$, where $n$ is an integer.
• Odd numbers: $\ldots, -7, -5, -3, -1, 1, 3, 5, 7, \ldots, 2n+ 1, \ldots$, where $n$ is an integer.
• Consecutive integers: $\ldots, n-1, n, n+1, n+2, n+3, \ldots$, where $n$ is an integer.

Number Line

• Consecutive integers: $\ldots, n-1, n, n+1, n+2, n+3, \ldots$, where $n$ is an integer.

Factors, Divisibility, and Remainders

Fractions and Decimals

• When dealing with fractions, one whole unit = 1.

Ratios, Proportions, and Percents

• A ratio is is a comparison between two quantities of the same kind. It can be expressed as $a$ to $b$, $a:b,$ $a/b$, or $\frac{a}{b}.$

• A proportion is an equation stating that two ratios are equivalent.

• A percent is a fraction of a hundred. It can be expressed as $\frac{a}{100},$ $a\%,$ or $a$ percent.

• The properties of proportion:
  If $a, b, c,$ and $d$ are nonzero and $\frac{a}{b}=\frac{c}{d},$ then $\begin{array}{r c l} \frac{b}{a} &=& \frac{d}{c}\\ \\ \frac{a}{c} &=& \frac{b}{d}\\ \\ ad&=&bc \\ \\ \frac{a}{a+b} &=& \frac{c}{c+d} \\ \\ \frac{a+c}{b+d}&=&\frac{a}{b}=\frac{c}{d}. \end{array}$

Sequences and Series

• Find the $n^\text{th}$ term in an arithmetic sequence: $a_n = a_1 + (n-1)d$.
• Find the $n^\text{th}$ term in a geometric sequence: $a_n = a_1 \times r^{n-1}$.
• Find the sum of $n$ consecutive terms in an arithmetic sequence: $S_n = \frac{n(a_1+a_n)}{2}$.
• Find the sum of $n$ consecutive terms in a geometric sequence: $S_{n}= a_1 \times \frac{r^n -1} {r-1}, \text{where}\ r\neq 1$.

Algebraic Manipulations

• If $a$ and $b$ are integers, and $a, a+1, a+2,\ldots, b-1, b$ is a sequence of consecutive integers, then the number of terms between $a$ and $b$ is $b-a+1$.

Polynomials

• $a^{2}-b^{2}=(a-b)(a+b)$
• $(a \pm b)^{2} = a^{2} \pm 2ab + b^{2}$

Exponents

• Rules of Exponents:

$\begin{array} { c | c } \text{ Rule name} & \text{ Rule } \\ \hline \text{Product Rule} & a^m \times a^n = a^{ m + n } \\ & a ^n \times b^n = (a \times b)^ n \\ \hline \text{Quotient Rule} & a^n / a^m = a^ { n - m } \\ & a^n / b^n = (a/b) ^ n \\ \hline \text{Negative Exponent} & a^ {-n} = \frac{1}{a^n} \\ \hline \text{Power Rule} & (a^n)^m = a^ { n \times m } \\ & a ^ { n^ m } = a ^ { \left ( n^ m \right) } \\ \hline \text{Fraction Rule} & a ^ { 1/n} = \sqrt[n]{a } \\ & \sqrt[m]{ a^n} = a^ { n/m} \\ \hline \text{Zero Rule} & a^0 = 1 \\ & 0^ a = 0 \text{ for } a > 0 \\ \hline \text{One Rule} & a^1 = a \\ & 1^a = 1 \\ \end{array}$

• The first few perfect squares are $(1, 4, 9, \ldots, 400),$ and cubes $(1, 8, 27, \ldots, 1000).$

• $\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$

Change the Subject

• Know the rules of exponents.

Inequalities

• $x^{2} \geq 0.$

• The properties of inequality:

  • If $a<b$, then $a+c<b+c$ and $a-c<b-c.$
  • If $a<b$ and $c<d$, then $a+c<b+d.$
  • If $a<b$ and $c>0$, then $ac<bc$ and $\frac{a}{c}<\frac{b}{c}.$
  • If $a<b$ and $c<0$, then $ac>bc$ and $\frac{a}{c} > \frac{b}{c}.$

• Reversing the inequality:

  • If $a<b$, then $-a>-b.$
  • If $a>b$, then $-a<-b.$
  • If $a$ and $b$ are both positive or both negative and $a<b$, then $\frac{1}{a}>\frac{1}{b}.$

• The properties of numbers between $0$ and $1$ $(\text{for } 0<a<1)$:

  • If $0 < b < 1$, then $ba<a$.
  • If $m$ and $n$ are integers such that $1<n<m$, then $x<x^{n}<x^{m}$.
  • $a<\sqrt{a}$.
  • $a<1<\frac{1}{a}$.

Absolute Value

• $|x| = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
• $x^{2} \geq 0.$

Functions

• The domain of $f(g(x))$ is the set of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.
• $\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
• For $f(x)=\sqrt{x},$ the domain is $x\geq 0$ and the range is $f(x) \geq 0.$

Linear Functions

• The slope of a line is defined as $\frac{(\text{change in}\ y)}{(\text{change in}\ x)}.$
• A line with a positive slope rises from left to right.
• A line with a negative slope falls from left to right.
• The slope-intercept form is $y=mx+b,$ where $m$ is the line's slope and $b$ is its $y$-intercept.
• The point-slope form is $y-y_{1}=m(x-x_{1}),$ where $m$ is the line's slope and $(x_{1}, y_{1})$ is a point on the line.
• The line $x = a$ is a vertical line that crosses the $x$-axis at $(a,0).$
• The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
• If two lines are parallel, their slopes are equal.
• If two lines are perpendicular, their slopes are negative reciprocals of each other.
• If two functions intersect at point $(x,y),$ then $f(x) = y = g(x)$.

Quadratic Functions

• The parabola $y=a(x-h)^{2}+k$ opens up if $a>0.$
• The parabola $y=a(x-h)^{2}+k$ opens down if $a<0.$
• The parabola $y=ax^{2}+bx+c$ opens up if $a>0.$
• The parabola $y=ax^{2}+bx+c$ opens down if $a<0.$
• The parabola $y=ax^{2}+bx+c$ has a $y$-intercept at $y=c.$
• The parabola $y=a(x-h)^{2}+k$ has a vertex at $(h, k).$
• The parabola $y=a(x-h)^{2}+k$ has an axis of symmetry at $x=h$.
• The parabola $y=ax^{2}+bx+c$ has a vertex at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right).$
• The parabola $y=ax^{2}+bx+c$ has an axis of symmetry at $x=-\frac{b}{2a}$.

Coordinate Geometry

• The line $x = a$ is a vertical line that crosses the $x$-axis at $(a,0).$
• The line $y = b$ is a horizontal line that crosses the $y$-axis at $(0,b).$
• If two lines are parallel, their slopes are equal.
• If two lines are perpendicular, their slopes are negative reciprocals of each other.
• If two functions intersect at point $(x,y),$ then $f(x) = y = g(x)$.
• Distance formula: $d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}.$
• Midpoint formula: $M=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right).$
• When transforming graphs, trace what happens to each point.

Functions as Models

• Exponential growth: $y=ba^{x}$, where $a>1$ and $b > 0.$
• Exponential decay: $y=ba^{x}$, where $0<a<1$ and $b>0.$

Newly Defined Functions

Direct and Inverse Variation

• Direct variation: $y=k\cdot x, \quad k\neq0.$
• Inverse variation: $y=k\cdot\frac{1}{x}, \quad k\neq0.$

Translating Word Problems

Word Problemes

• (Distance) = (Rate) $\times$ (Time).

Student-Produced Response

Lines and Angles

• Acute: angles with measure $< 90^\circ$.
• Right: angles with measure $= 90^\circ$.
• Obtuse: angles with measure $> 90^\circ$ and $< 180 ^ \circ$.

• Straight: angles with measure $= 180^ \circ$.

• Angles at a point sum to $360 ^ \circ$.

• Angles on a line sum to $180 ^ \circ$.

• Vertical angles are equal.

• $\angle A$ and $\angle B$ are complementary if $m\angle A + m\angle B=90^\circ.$

• $\angle A$ and $\angle B$ are supplementary if $m\angle A + m\angle B=180^\circ.$

• Segment Addition Postulate: If point $B$ lies between points $A$ and $C,$ then $AB + BC = AC.$

• Angle Addition Postulate: If point $B$ lies inside $\angle AOC,$ then $\angle AOB + \angle BOC = \angle AOC.$

• The midpoint divides a segment into two congruent segments. If $M$ is the midpoint of $\overline{AB},$ then $\overline{AM} \cong \overline{MB}, AM = MB = \frac{1}{2}AB.$

• The angle bisector is a ray or a line that divides an angle into two congruent angles. If $\overrightarrow{BD}$ is the bisector of $\angle ABC,$ then $\angle ABD = \angle DBC = \frac{1}{2} \angle ABC.$

• Angles in a triangle sum to $180 ^ \circ$.

Parallel Lines

• The Properties of Parallel Lines:
  Consider the two parallel lines $m$ and $n,$ and the transversal, $p.$
  

  • Interior angles are angles located on the inside of the parallel lines: $\angle 3, \angle 4, \angle 5,\ \text {and}\ \angle 6.$

  • Exterior angles are angles located on the outside of the parallel lines: $\angle 1, \angle 2, \angle 7,\ \text{and}\ \angle 8.$

  • Corresponding angles are two angles located in the same position relative to the parallel lines. Corresponding angles are congruent: $\angle 1 \cong \angle 5, \quad \angle 2 \cong \angle 6, \quad \angle 3 \cong \angle 7, \quad \angle 4 \cong \angle 8.$

  • Alternate-interior angles are two nonadjacent interior angles located on opposite sides of the transversal. Alternate-interior angles are congruent: $\angle 3 \cong \angle 5 \quad \text{and} \quad \angle 4 \cong \angle 6.$
  • Alternate-exterior angles are two nonadjacent exterior angles located on opposite sides of the transversal. Alternate-exterior angles are congruent: $\angle 1 \cong \angle 7 \quad \text{and} \quad \angle 2 \cong \angle 8.$
  • Same-side interior angles are two nonadjacent interior angles located on the same side of the transversal. Same-side interior angles are supplementary: $\angle 3 + \angle 6 = 180^\circ \quad \text{and} \quad \angle 4 + \angle 5 = 180^\circ.$
  • Same-side exterior angles are two nonadjacent exterior angles located on the same side of the transversal: $\angle 1 + \angle 8 = 180^\circ \quad\text{and} \quad \angle 2 + \angle 7 = 180^\circ.$

Triangles

• Scalene triangle: none of its sides are congruent.
• Isosceles triangle: at least two sides are congruent.
• Equilateral triangle: all three sides are congruent.
• Acute triangle: All angles measure less than $90^\circ$.
• Obtuse triangle: One angle measures more than $90^\circ$ and less than $180^\circ$.
• Right triangle: One angle measures $90^\circ.$

• Isosceles Triangle Theorem: The angles opposite the two congruent sides are congruent.

• The three angles of an equilateral triangle are all equal to $60^\circ$.

• The measures of the angles in a triangle add to $180^\circ.$

• The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles.

Inequalities:

• If one side of a triangle is longer than another side, then the angle opposite the first side is bigger than the angle opposite the second side.

• If one angle in a triangle is bigger than another angle, then the side opposite the first angle is longer than the side opposite the second angle.

• Triangle Inequality: The sum of the lengths of any two sides in a triangle is greater than the length of its third side.

Perimeter and Area:

• Perimeter of a polygon is defined as the sum of the lengths of its sides.

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$

• Area Addition Postulate: The area of a region is the sum of the areas of its non-overlapping parts.

• If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.

• If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights.

• If two figures are similar, and their scale factor is $a:b,$ then the ratio of their perimeters is $a:b$ and the ratio of their areas is $a^2:b^2.$

Right Triangles

• Pythagorean Theorem
  In a right triangle, the square of the hypotenuse equals the sum of the square of the legs: $c^2 = a^2 + b^2.$
• If $c^2 = a^2 + b^2,$ then $\angle C = 90^\circ$ and $\triangle ABC$ is right.
• If $c^2 < a^2 + b^2,$ then $\angle C < 90^\circ$ and $\triangle ABC$ is acute.

• If $c^2 > a^2 + b^2,$ then $\angle C > 90^\circ$ and $\triangle ABC$ is obtuse.

• $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle
  In a $30^\circ\text{-}60^\circ\text{-}90^\circ$ triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is $\sqrt{3}$ times as long as the shorter leg.

• $45^\circ\text{-}45^\circ\text{-}90^\circ$ Triangle
  In a $45^\circ\text{-}45^\circ\text{-}90^\circ$ triangle, the hypotenuse is $\sqrt{2}$ times longer than either of the legs.

Polygons

• $A_\text{parallelogram} = bh,$ where $b$ is the length of the base, and $h$ is the height.
• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Area of a square with side length $s: A_{\square} = s^2.$
• The sum of the measures of the interior angles of a convex polygon with $n$ sides is $180(n-2).$
• The sum of the measures of the exterior angles, one per vertex, of any convex polygon is $360^\circ.$

Circles

• The circumference of a circle with radius $r$ and diameter $d: C = 2\pi r = \pi d.$
• Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
• The measure of an arc equals the measure of its central angle.
• The length of an arc with measure $x$ is $\frac{x}{360}\cdot 2 \pi r.$
• The area of the sector formed by an arc measuring $x$ and two radii is $\frac{x}{360} \cdot \pi r^2.$

Congruence and Similarity

• If two figures are similar, and their scale factor is $a:b,$ then the ratio of their perimeters is $a:b$ and the ratio of their areas is $a^2:b^2$.

Composite Figures

Solid Geometry

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ\text{-}60^\circ\text{-}90^\circ$ and the $45^\circ\text{-}45^\circ\text{-}90^\circ$ theorems.
• Area of a circle with radius $r: A_{\bigodot} = \pi r^2.$
• The perimeter of a square with side length $s$: $P_{\square} = 4s.$
• The volume of a cube with edge length $s$: $V = s^3.$
• The volume of a rectangular solid with length $l,$ width $w,$ and height $h: V = l \cdot w \cdot h.$
• The surface area of a cube with edge length $s$: $SA = 6s^2.$
• Volume of a cylinder with base radius $r$ and height $h: V = \pi r^2 h.$

Conceptual Geometry

Mean, Median, and Mode

• The average of $n$ numbers is the sum of the numbers divided by $n.$
• If the average of a set of numbers is $A$ and a new number $x=A$ is introduced to the set, the new average will also equal $A.$
• If $n$ numbers are arranged in increasing order, the median is the middle value if $n$ is odd, and it is the average of the two middle values if $n$ is even.
• In a set of numbers, the mode is the number that appears most frequently.
• To find the weighted mean of some numbers, find the product of each number and its weight, then divide the sum of these products by the sum of the weights.

Data - Tables

Data - Graphs and Charts

Sets and Venn Diagrams

• The union of two sets, $A$ and $B,$ is that collection of elements that are in $A,$ or in $B,$ or in both $A$ and $B.$
• The intersection of two sets, $A$ and $B,$ is that collection of elements that are only in both $A$ and $B.$
• If every element in set $A$ is an element in set $B,$ then $A$ is a subset of $B.$

Counting and Probability

• If $a<b$ are two integers, the number of integers between $a$ and $b$ when one endpoint is included is $b-a.$
• If $a<b$ are two integers, the number of integers between $a$ and $b$ when both endpoints are included is $b-a+1.$
• If $a<b$ are two integers, the number of integers between $a$ and $b,$ endpoints NOT included is $b-a-1.$
• If there are $n$ ways for an event to happen and $m$ ways for another event to happen, then the number of ways for both events to happen is $m\cdot n.$
• If $P(A)$ is the probability that event $A$ will occur, then $0 \leq P(A) \leq 1.$
• If $P(B)$ is the probability that event $A$ does NOT occur, then $P(A) = 1- P(B).$
• Assuming that all the possible outcomes of an event $A$ are equally likely, the probability that $A$ will occur is $P(A) = \frac{(\text{\# of favorable outcomes})}{(\text{total \# of outcomes})}.$
• Two events are independent if the outcome of one does not affect the outcome of the other.
• If events $A$ and $B$ are independent, then $P(A\ \text{and}\ B) = P(A) \cdot P(B).$
• The probability that two events, $A$ and $B,$ happen together is $P(A\ \text{and}\ B) = P(A)\cdot P(B\ \text{given}\ A).$
• If events $A$ and $B$ are mutually exclusive, then $P(A\ \text{or}\ B) = P(A) + P(B).$
• If a point is chosen at random in a geometric figure, the probability that the point lies in a particular region is: $\frac{(\text{area of region})}{(\text{area of whole figure})}.$