SAT 101 Flashcards
- slope-intercept form of a line
arithmetic and algebra
y=mx+b
- Vertex Form of a Parabola/Quadriatic
arithmetic and algebra
y = a(x-h)² + k
be able to recognize this vertex form and convert quadratics to this form. The values of hand k give you the coordinates of the vertex, (h,k)
- Distance formula-derived from the Pythagorean Theorem and is useful for quickly finding the distance between two points.
arithmetic and algebra
_________________________________
d = √ ( x₂ - x₁)² + (y₂ - y₁) ²
Take the values of the coordinates and plug them into this formula to find the distance, and be sure to apply the squares and the square root at the right step.
- Quadratic Formula
arithmetic and algebra
\_\_\_\_\_\_\_\_\_\_\_\_\_
x = - b +- √ b² - 4ac
\_\_\_\_\_\_\_\_\_\_\_\_\_
2a
helps you find the roots of a quadratic equation (parabola) if you can’t easily factor it. You need the quadratic to be in the form y= ax2 + bx + c, and then yo simply plug the coefficents and constants into the formula. Note that you will get two answers because there is a plus and the minus sign in the numerator.
- Exponent Rule (Multiplication)
arithmetic and algebra
aⁿaᵐ + a ⁿ+ᵐ
knowing how to manipulate exponents in a variety of ways will help you tremendously. If you have the ** same base number** raised to different power multiplied together you can add the exponents together.
- Exponent Rule (Division)
arithmetic and algebra
aᵐ
__ = a ᵐ-ⁿ
aⁿ
if you have the same base number raised to different powers being divided, you can subtract the exponents. You can also rewrite the expression on the right to mirror the one on the left.arithmetic and algebra
- Exponent Rule (Power Raised to a Power)
arithmetic and algebra
(aⁿ) ᵐ = a ⁿ . ᵐ
Raising a power to another power is the same as multiplying the exponents together. If you don’t remember these, brush up.
- Binomial Product 1 - Difference of Squares
arithmetic and algebra
(x-y) (x+y) = x² - y²
The best times to recognize teh binomial products ad quickly factor them is on no calculator section. You don’t have to FOIL or use any other method–you can quickly convert fro the factored form to the expanded form on sight. The difference of squares is used often by the SAT makers in a variety of contexts.
- Binomial Product 2 - Perfect Squares Trinomial (Positive)
arithmetic and algebra
(x + y)² = x² + 2xy + y²
This is a good one to recognize. It saves you time, but it’s a little more difficult to catch than the expanded difference of squares. A good way to know if you’re dealing with one is to look at the first and last values–are they perfect squares?
- Binomial Product 2 - Perfect Squares Trinomial (Negative)
(x - y)² = x² - 2xy + y²
While the factored form doesn’t involve coefficents, the binomial products on the SAT often do. Practice recognizing these patterns by inputting coefficents in front of x and constants for y on the eft-hand side. Then multiply out the expression to see how the pattern works with different combinations.
- Complex Conjugate
arithmetic and algebra
(a + bi) ( a - bi) = a² + b²
Most SAT will have at least one question that involves manipulating imaginary numbers. The complex conjugate allows you to get rid of the imaginary part of a complex number and leaves you with a real number (notice how it resembles the difference of squares!).
When given a complex number in the form of a + bi, the conjugate is a - bi
- Exponential Growth and Decay
arithmetic and algebra
y = a ( 1 +- r) ˣ
This will help on several SAT questions, as you may need to interpret or manipulate these equations. The value a is the initial value, r is the rate of growth when positive, rate of decay when negative.
- Simple Interest
Rates, percentages and statistics
A = P r t
P is principle amount, r is the interest rate as expressed as a decimal, and t is for time, usually in years.
- Compound Interest
Rates, percentages and statistics
A = P (1 + r/n) ⁿᵗ
The n represents the number of times that the interest is compounded during 1 t. For example, if the interest is compounded quarterly over the course of the year, then n = 4.
- Average/Mean
Rates, percentages and statistics
In math, the words average and mean are the same thing: the number you get when you take the sum of a set and divide it by the number of values in the set.
Make sure to understand difference between mean and median.
- Random Sampling
Rates, percentages and statistics
This isn’t technically a formula, but many of the statistics-based problems on the SAT focus more on interpreting concepts in context rather than performing math operations. Random sampling is when you select participants for a study at random within your population. It ensures that your study is representative of the population.
- Random assignment
Rates, percentages and statistics
Random assignment is when the participants in a study are assigned a treatment or trial at random. It reduces bias in your study, and means that you can attribute causation in regards to the treatment. On the SAT, you’re often asked about what will reduce bias or how much you can generalize results to the rest of the population. In these instances, you need to identify random sampling and random assignment.
- Standard Deviation
Rates, percentages and statistics
You won’t need to calculate standard deviation for the SAT, but you will be tested on it conceptually, as with random sampling and random assignment. Standard deviation is the measure of spread in the data set. A higher standard deviation means greater spread, and lower standard deviations mean smaller spread. You’ll need to know how changes in the data set might affect the standard deviation by making it greater or smaller.
- Area of an Equilateral Triangle
Geometry and Trigonometry
A = √ 3s²
_____
4
The regular area of a triangle formula is provided on the SAT reference sheet, but it requires that you know the height. Sometimes you aren’t given the height, so you’ll need to calculate it, but you can quickly find the area of an equilateral by plugging the length of one of its sides into the formula above. No need to calculate the height!
- Equation of a Circle
Geometry and Trigonometry
(x - h ) ² + (y - k) ² = r²
There is usually one question involving the equation of a circle. In this equation, (h,k) is the coordinate for the center of the circle, and r is the radius of the circle.
- Sine Ratio
Geometry and Trigonometry
Some students get nervous when they hear that trig is on the SAT, but it most often appears in the form of trig ratios. Remember that **for a given angle in a right triangle, the value of sine is the **length of the opposite side divided by the length of the hypotenuse, or opposite/hypotenuse.
- Cosine Ratio
Geometry and Trigonometry
Just like with sine, remember what the cosine ratio is:
the length of the adjacent side divided by the length of the hypotenuse, or adjacent/hypotenuse.
- Tangent Ratio
Geometry and Trigonometry
Last but not least, the tangent ratio is the length of the opposite side divided by the length of the adjacent side, or opposite/adjacent. Some students find the menmonic SOH CAH TOA helpful for remembering trig ratios.
- Degrees to Radians
Geometry and Trigonometry
While the most common form of trig are the basic ratios, you may encounter things like the unit circle or more advanced math. If you need to convert degrees to radians, multiply the degrees by π/180. If you need to convert radians to degrees, multiply the radians by 180/π.
- Pythagorean Theorem
Geometry and Trigonometry
a² + b² = c²
The Pythagorean Theorem applies to right triangles, and allows you to solve for one of the side lengths given any other side length. a and be are the legs of the triangle, and c is the hypotenuse.
- Regular Polygon Interior Angle.
The SAT will probably involve one question with a regular polygon that isn’t a triangle or square. Regular polygons have unique and consistent properties based on their number of sides, and knowing these properties can help you solve these problems. This equation tells you what the degree measure at each angle is based on the the number of sides of n.
Geometry and Trigonometry
(n - 2) 180
_______
n
- 3-4-5 Triangle
Geometry and Trigonometry
The SAT provides you with two special right triangles on your reference sheet–30-60-90 and 45-45-90 triangles. However, the 3-4-5 is a special right triangles with sides that are straightforward integers.
This triangle is often incorporated into SAT problems, especially the no-calculator portion, so be on the look out for it! It can save you having to use the Pythagorean theorem.
- 5 -12-13 triangle
Geometry and Trigonometry
Another special right triangle with whole number sides, the 5-12-13 is less well-known and shows up less often than 3-4-5. Still it helps to be able to quickly solve the remaining sides without the Pythagorean theorem, so check for these numbers or their multiples in triangle positions!
- Length of Arc in a Circle
Geometry and Trigonometry
length of arc = central angle
___________ π d
360
you may find a question either about arc or sectors of a circle. An arc is the length between two points on a circle, usually measured by extending two radii from the center of the circle with an angle formed between them. You can use the degree measure of the arc as a fraction of 360 and multiply it by the equation for the circumference to find the length of the arc.
- Area of Sector in a Circle
Geometry and Trigonometry
area of sector = central angle
____________ π r ²
360
Like an arc, the sector is the area in between two radii extending from the circle, sort of like a slice of pie. Again, multiply the degree measure as a fraction of 360 and multiply it by the equation for the area of a circle to find the area of the sector.
It’s up to you to determine what formulas to apply! Practice using them!
Practice using these formulas–it is up to you to determine what formulas apply. When you practice using formulas with a variety of problems, you’ll be able to quickly identify which formula to use.
Bonus tip: memorize the perfect squares and perfect cubes. This can help you with quadratic equations that involve squares, and the cubes are often used in solving problems with exponents.
1 x+x x²
11² = 1 + 1+1 + 1²
= 121
121
11²
144
12²
169
similar to 196
13²
196
four short of two hundred
14²
4 short of 200
225
15²
squares of numbers ending in 5 ALWAYS end in 25
256
16²
2⁸
Important in IT because BYTE is 256 ( this square starts with 25, where previous square 15² ended (225)
17²
289
7 teen squared….to . 8,9
18²
324
32 is multiplication of 8 x 4
24 is also a multiple of 8
remember by 8 x3 is 24
361
19²
memorize it , but if you rotate 19 you get 61.
400
20²
21²
441
44
22²
484
44
4 4 FOURS EVERYWHERE
2 2s squared is 4 4’s
23²
529
DEAL WITH
2 3²
24²
576
24 short of 600
25²
625
25 above 600/always ends with 25
1²
1
2²
4
3²
9
4²
16
5²
25
6²
36
7²
49
8²
64
8 6 4
9²
81
9= 8+ 1
10²
100
1
1³
8
2³
27
3³
64
4³
125
5³
216
6³
343
7³
512
8³
729
9³
1000
10³
1331
11³
1728
12³
2197
13³
2744
14³
