SAT General Tips

The following tips are useful when working on SAT problems.

• Follow the order of operations.
• Read the entire question carefully.
• The simplest choice may not be the correct one.
• The complicated choice may not be the correct one.
• Look for short-cuts.
• If you can, verify your choice.
• Just because a number appears in the question doesn't mean it is the answer.
• Plug and check.
• Identify irrelevant information.
• Eliminate obviously wrong answers.
• Select the answer with the correct sign!
• When distributing, be careful with signs!
• Avoid long solutions.
• Identify irrelevant information.
• Read the answers carefully.
• Use a calculator.
• Replace variables with numbers.
• Be careful with signs!
• Pay attention to units.
• Look for a counter-example.
• Use reasoning skills.
• Follow directions exactly.
• If a diagram is drawn to scale, trust it.

Numbers

• Know the properties of even and odd 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.
• Only assume that the tick marks are equally spaced, nothing more.

Factors, Divisibility, and Remainders

Fractions and Decimals

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

Ratios, Proportions, and Percents

• Know the properties of proportions.
• Pay attention to units.

Sequences and Series

• For an arithmetic sequence, $a_{n}=a_{1}+(n-1)d$.
• For a geometric sequence, $a_n = a_1 \times r^{n-1}$.
• For an arithmetic series, $S_n = \frac{n(a_1+a_n)}{2}$.
• For a geometric series, $S_{n}= a_1 \times \frac{r^n -1} {r-1}, \text{where}\ r\neq 1$.

Algebraic Manipulations

• Follow the order of operations.

Polynomials

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

Exponents

• Know the rules of exponents.
• Recognize first few perfect squares (1, 4, 9, ..., 400) and cubes (1, 8, 27, ..., 1000).
• The square of a number is always positive.
• $\sqrt{x^{2}} = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$

Change the Subject

• Know the rules of exponents.

Inequalities

• $x^{2} \geq 0.$
• Know the properties of inequality.
• Multiplying (or dividing) both sides of an inequality by a negative number reverses its sign.
• Know the properties of numbers between $0$ and $1$.

Absolute Value

• $|x| = \begin{cases} -x &\mbox{if } x < 0 \\ x & \mbox{if } x \geq 0. \\ \end{cases}$
• $x^{2} \geq 0.$
• Know the properties of inequality.
• Multiplying (or dividing) both sides of an inequality by a negative number reverses its sign.

Functions

• Don't switch the $x$- and $y$-coordinates of a point.
• 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.
• Slope-intercept form: $y=mx+b,$ where $m$ is the line's slope, and $b$ its $y$-intercept.
• Point-slope form: $y-y_{1}=m(x-x_{1}),$ 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 as $(a,0).$
• 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 $f(x) = y = g(x)$.
• Don't switch the $x$- and $y$-coordinates of a point.
• When transforming graphs, trace what happens to each point.

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 $f(x) = y = g(x)$.
• Don't switch the $x$- and $y$-coordinates of a point.
• 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

• Follow directions exactly.

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

• Angles at a point sum to $360^\circ.$
• Angles on a line sum to $180^\circ.$
• $\angle A$ and $\angle B$ are complementary if $\angle A + \angle B=90^\circ.$
• $\angle A$ and $\angle B$ are supplementary if $\angle A + \angle B=180^\circ.$
• Vertical angles are congruent.
• The angle bisector divides an angle in half.
• The midpoint of a segment divides it in half.
• If a diagram is drawn to scale, trust it.

Parallel Lines

• Know the properties of parallel lines.
• Angles on a line sum to $180^\circ.$
• $\angle A$ and $\angle B$ are complementary if $\angle A + \angle B=90^\circ.$
• $\angle A$ and $\angle B$ are supplementary if $\angle A + \angle B=180^\circ.$
• Vertical angles are congruent.
• The angle bisector divides an angle in half.
• Angles in a triangle sum to $180^\circ.$
• The two acute angles in a right triangle are complementary.
• An exterior angle in a triangle equals the sum of the two nonadjacent interior angles.
• If a diagram is drawn to scale, trust it.

Triangles

• The angles opposite the two congruent sides in an isosceles triangle are congruent.
• 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.
• 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.
• The perimeter of a polygon equals the sum of the lengths of its sides.
• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• 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: $a^2 + b^2 = c^2.$
• If $c^2 = a^2 + b^2,$ then $m\angle C = 90$ and $\triangle ABC$ is right .
• If $c^2 < a^2 + b^2,$ then $m\angle C < 90$ and $\triangle ABC$ is acute.
• If $c^2 > a^2 + b^2,$ then $m\angle C > 90$ and $\triangle ABC$ is obtuse.
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ theorems.
• AA postulate: two triangles are similar if two angles of one triangle are congruent to two angles of the other triangle.
• The measures of the angles in a triangle add to $180^\circ.$
• The perimeter of a polygon equals the sum of the lengths of its sides.

Polygons

• Know the properties of parallelograms.
• $A_\text{parallelogram} = bh,$ 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 $\frac{x}{360} \cdot \pi r^2.$

Solid Geometry

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ 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 $h: V = \pi r^2 h.$

Composite Figures

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ theorems.
• The perimeter of a square with side length $s$: $P_{\square} = 4s.$
• Area of a square with side length $s: A_{\square} = s^2.$
• Area of a rectangle with length $l$ and width $w: A = l\cdot w.$
• 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 $h: V = \pi r^2 h.$
• 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 $x^\circ$ is $\frac{x}{360}\cdot 2 \pi r.$
• The area of the sector formed by an arc measuring $x$ and two radii is $\frac{x}{360} \cdot \pi r^2.$

Solid Geometry

• Area of a triangle with height $h$ and base $b$: $A_{\triangle} = \frac{1}{2}bh.$
• Know the $30^\circ-60^\circ-90^\circ$ and the $45^\circ-45^\circ-90^\circ$ 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 $h: V = \pi r^2 h.$

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 the 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 the 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 $b$ when the endpoints are 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 $P(A\ \text{and}\ B) = P(A) \cdot P(B).$
• 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: $\frac{(\text{area of region})}{(\text{area of whole figure})}.$