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: \(\frac{(\text{area of region})}{(\text{area of whole figure})}.\)