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