Operations
One Variable
– Like terms: Terms whose variables (such as x or y) with any exponents (such as the 2 in x2) are the same. Examples: 7x and 2x are like terms because they are both “x”.
– REMINDER: Always check what they are asking for before selecting your answer. For example, the questions may ask “what is x + 4” as opposed to just “what is x”.
Two or More Variables
– 3 Quadratic Identities (unfactored to factored form)
(x2-y2)=(x+y)(x-y)
x2+2xy+y2=(x+y)2
x2-2xy+y2=(x-y)2
Variables in Choice
System of Equations
METHOD 1: Substitution
2x + 4y = 6 AND y = 5 —–> 2x + 4(5) = 6
METHOD 2: Elimination
2x + 4y = 6
Subtract these two equations to eliminate the “x”:
2x – y = – 9
5 y = 15
y = 3
METHOD 3: Set Equal
y = 2x + 3 AND y = 4x + 1 ——> 2x + 3 = 4x + 1
Absolute Value
Break the equation into two equations (positive and negative)
Absolute Value Equations will always have two answers!
| 2x + 4 | = 6
2x + 4 = + 6 2x + 4 = – 6
2x = 2 2x = – 10
x = 1 x = -5
Inequalities
– Pretend the inequality sign is an equal sign and solve for the variable!
– When you divide or multiply by a negative, FLIP THE SIGN!
-6x – 9 > 3
Add 9
-6x > 12
Divide by -6 AND flip the sign
x < -2
Exponents
Functions
– The “x” is Just a Place-Holder! It is just there to show us where the input goes and what happens to it. It could be anything!
Sequences
– Sequences are all about finding the pattern!
Quadratics
ax^2 + bx + c = 0
METHOD 1: Factoring
– Find two numbers that multiply to the last term and add to the middle term
METHOD 2: Quadratic Formula
The Discriminant
– The discriminant is everything under the radical in the quadratic equation = b^2 – 4ac
– If the discriminant is POSITIVE, then there are 2 real roots (“roots” is another word for “solutions” when equations are written in ax^2+by+c = 0 form).
– If the discriminant is ZERO, then there is 1 real root.
– If the discriminant is NEGATIVE, then there are no real roots. (#13 C Test 6)
– Know how to complete the square
Defined Operations
Imaginary Numbers
– Binomial addition involving constants and i by combining like terms (adding and subtracting complex numbers)
– Multiplying by the conjugate of the denominator with complex numbers (#11 Test 2)
Number Theory
– Integers: whole numbers, including zero and negative whole numbers
– Prime Numbers: positive integers that are only divisible by themselves and the number 1 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, etc…
Divisibility & Remainders
– Polynomial Remainder Theorem (#29 C Test 1) (#7 NC Test 3)
– Divisible by 2 = last digit 0, 2, 4, 6, 8?
– Divisible by 3 = sum of digits divisible by 3?
– Divisible by 4 = last 2 digits divisible by 4?
– Divisible by 5 = last digit 0 or 5?
– Divisible by 6 = follow rule 2 and 3?
– Divisible by 8 = last three digits divisible by 8?
– Divisible by 9 = sum of the digits divisible by 9?
– Divisible by 10 = last digit 0?
Ratio & Proportions
– Direct and Indirect Proportion (a1/b1)=(a2/b2) and (a1a2 = b1b2), respectively
Percents
– Percentage = (Part/Whole)
– Percent Change = (Difference/Original) x 100)
Triangles
– Special Triangles (3/4/5, 5/12/13, 30/60/90, 45/45/90)
– Pythagorean theorem is used to find the third side of a right triangle. It is ONLY for right triangles. NOTE: Hypotenuse is always the longest side.
– The largest side is opposite the largest angle. The smallest side is opposite the smallest angle.
– All three angles add up to 180 degrees!
– The sides of similar triangles all have the same respective proportions. (#17 NC Test 1, #18 NC Test 2)
– The Third Side Rule for Triangles (a-b) < c < (a+b) if c represents the “third side” and b and a represent the lengths of the other two sides.
– The proportion of distance that you travel along the hypotenuse of a right triangle is equal to the proportion of distance that you travel along both legs. (#16 NC Test 4)
Circles & Sectors
– The Circle Proportionality Formula (Slice/Area = Arc/Circumference = Measure of Inner Angle/360)
– The arc measure formed by an angle with its vertex on a circle is double the measure of the angle. (#36 C Test 5)
– The equation of a circle with center (h,k) and radius r: (x-h)2 + (y-k)2= r2 (#24 C Test 1)
2D & 3D Objects
Surface Area of a Cube = 6s2
Area of a triangle = 1/2 ab sin C
– Unfortunately all the formulas below are not included on the ACT, unlike on the SAT.
– The vertex of a parabola is located at the midpoint of its x-intercepts (#12 NC Test 3)
– The vertex (h,k) form of a parabola: a(x-h)^2 + k
– When an upward projectile reaches its highest point, its velocity is zero.
– When an upward projectile lands, its height is zero.
– To find the intersections of two lines, set them equal to one another (#13 test 4)
– In a system of linear equations, there is no solution if the slopes of the two lines are the same (parallel) and the y-intercept is different. (see #9 Test 3) Conversely, there are infinitely many solutions is the slopes of the two lines are the same and the y-intercept is also the same (#20 NC Test 2)
– The “zeroes” or “roots” of a function are the x-coordinates where it crosses the x-axis (and where the y value outputs zero).
– A polynomial of Nth degree has at most N-1 changes in direction.
– Domain and Range
– Parallel Lines and Transversals (#36 Test 1)
– The Formula for a Line (standard y=mx+b format as well as point-slope format: y-y1 = m(x-x1), and the slope equation (y2-y1) / (x2-x1) ).
– Positive and Negative Associations in Graphs (#5 C Test #1)
Number Lines & Line Segments
A line extends forever in both directions.
A line segment has two endpoints.
A ray is a part of a line that has one endpoint and goes on infinitely in only one direction.
A midpoint is the middle of the line. To solve for the midpoint, take the average of the x value and the average of the y value.
Angles
A line measures 180degrees.
Types of angles to know:
– opposite angles
– supplementary angles
– opposite interior angles
– complimentary angles
Degrees & Radians
– π radians = 180 degrees (#19 NC Test 2)
Probability
Probability = (Desired Possibilities / Total Possibilities)
Permutations & Combinations
Permutation = arranging the objects or numbers in order
Combinations = selecting objects or numbers from a group, in which order doesn’t matter.
Simple & Compound Interest
Simple interest is based on the principal amount of a loan or deposit. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period.
Rates & Work
Distance = Rate x Time (#38 C Test 5, #9 C Test 3)
Mean, Median, & Mode
Mean: average = (Total / Number of things)
Median: middle number
Mode: most frequent number
Concept: Weighted Averages (#19 NC Test 5)
Standard Deviation & Range
Tables & Charts
– Box and whisker plots (showed up on March 2018 SAT)
Matrices
Scientific Notation
– Negative exponents move the decimal to the left.
– Positive exponents move the decimal to the right.
Word Problems
– Word problems can come in the form of any topic above. The best trick is to convert the words into a mathematical equation first!
Other Helpful Study Materials
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