SAT Math

percentage change

(difference/original) X 100

% change

(new value - old value) / old value all X 100

midpoint formula

(x₁+x₂)/2, (y₁+y₂)/2

sum of the roots

-(b/a)

i^2

-1

solve for x. -100n^2 +1000n = 0

-100n(n-10) = 0

(-3a -3) / a+1 =

-3. not -6. Why? -3(a+1) / a+1. Exactly. Always factor out when you do problems like this.

sum of roots

-b/a

i^3

-i

i^4

1

what to do when you are stuck?

1. Reread question 2. Relate to topic and think about everything you know in that topic and how it may relate to question 3. plug in if possible

pi radians in degrees

180

sum of the degrees in an n-sided polygon

180(n-2)

sum of the interior angles of a polygon

180(n-2)

triangle ratios to memorise

3-4-5, 6-8-10, 5-12-13, 7-24-25, 8-15-17

triangle ratios

3-4-5,6-9-10,12-16-20,15-20-25,5-12-13,10-24-26,15-36-39

Convert 45 degrees to radians

45 X (Pi radians / 180) = Pi/4 radians

-5/3 =?

5/-3

special triangle with xroot3

90-60-30, x to the left, xroot3 bottom next to 30 degrees, 2x hypotenuse

semiannual compound interest formula

A = initial amount(1+interest rate/number of times interest is compounded per year) ^number of times compounded per year X number of years

when is a function undefined?

A function is undefined when the denominator is equal to zero

ratio of AC to BC

AC/BC

what to do when you have a long equation like x^6 -9x^2 -8x^4

BREAK IT APART. x^2(x+3)(x-3)(x^2+1) = 0

what to do when you see a fraction?

Cancel it out immediately

the function f is graphed in the xy-plane above. If the function g is defined by g(x)=f(x)+4, what is the x-intercept of g(x)?

Firstly put it into y=mx+c form. f(x) is y=-2x+2. If g(x) is 4 more than f(x), then g(x) is equal to y=-2x+6. Hence x=3. Make sure to always put into y=mx+c form when solving a problem like this and remember that x-intercept means y=0!!!!

relationship between the sum of two sides of a triangle and the third

For any triangle, the sum of any two sides must be greater than the third

A group of 30 students order lunch from a restaurant. Each student gets either a burger or a salad. The price of a burger is $5 and the price of a salad is $6. If the group spent a total of $162, how many students ordered burgers?

Let x be the number of students who ordered burgers and y be the number who ordered salads. x+y=30. 5x+6y=162. Simultaneous equations type beat. 18 students ordered burgers.

1/x + 1/y = 1/p. Solve for x.

Multiple both sides by xyp

(root4x -y) squared

NOT 4x-y^2. its (root4x-y)(root4x-y)

the function f(x)=x^3+1 is graphed in the xy-plane above. If the function g is defined by g(x)=x+k, where k is a constant, and f(x)=g(x) has 3 solutions, which of the following could be the value of k?

Plug in the different answer options to k and draw the new lines accordingly. Whichever one intersects 3 times is correct.

how do we find function f undefined?

Set the denominator to zero

lets say we have the line 2x+3y=12. Find the x-intercept.

Set y to 0. 2x+3(0)=12. x=6.

special triangle with two Xs

Sides X, X, xroot2 on hypotenuse, 90, 45, 45

if x=5 leads to f(x) being undefined, what is the domain of f(x)

The domain of f is all real numbers except for 5

how many zeroes does the function f have?

The graph cross the x-axis three times, so three distinct zeroes.

Domain

The set of all possible input values(x) to a function (values that don't lead to an invalid operation or an undefined output)

degrees to radians

X (Pi radians / 180)

radians to degrees

X 180/Pi radians

(a/b)/c

a/bc

if the system of equations have no solutions, what is the value of a: -ax -12y =15, 4x+3y=-2

a=16

a/(b/c)

ac/b

what can you deduce from f(x)>0

all the y values are greater than 0. Hence the line never goes below the x-axis.

discriminant

b^2 -4ac (everything under the root in quadratic equation)

product of roots

c/a

product of the roots

c/a

x intercept of the line (Ax + By = C) form

c/a

y intercept of the line (Ax + By = C form)

c/b

how to get a quadratic into vertex form?

complete the square. make sure that the coefficient of x^2 is 1 or take the average of the roots to get the x-coordinate. Then plug that value into the quadratic to get the y-coordinate.

inverse proportionality

d=K/s

direct proportionality

d=Ks

Dividend, quotient, divisor, remainder

dividend=quotient x divisor + remainder

is greater than on a graph line or dotted?

dotted

always check for what when factoring?

extraneous solutions

final amount in growth or decay

final amount = original amount (1 +or- rate) ^number of changes

if a question asks anything related to how many real solutions etc what do you do?

find the discriminant

f(x) - c

graph shifts down by c units

what do you know if the remainder is 0?

if the remainder is 0, x-2 is a factor of x^2-3x+2. this is demonstrated by (x-2)(x-1)

how can you calculate exterior angle

it is the sum of the 2 interior angles furthest from it

what is a horizontal asymptote?

it is when the graph never goes below the x-axis. its another line that the graph approaches but never crosses.

f(x+c)

left c units

is greater than or equal to line or dotted?

line

what to do when an equation is convoluted eg (x+1)^2 + 5(x+1) -24 = 0

make (x+1) one unit of A. Now all you have is A^2 + 5A -24 = 0

n^2 = 9. what does n equal?

n=3, n=-3

Part over whole of circle

part/whole = central angle/360 = arc length/2PiR = sector area/PiR^2

which of the following could be the graph of y=x^3 +2x^2+x+1

plug in x=0. Hence y=1. Find the graph with the point (0,1)

With what type of numbers does remainder theorem work?

remainder theorem only works with monomials like x+1. If we were dividing x^3 +1 by something like x^2 +2, we would have to use synthetic division.

Does a negative number raised to an odd power go positive or negative?

remains negative

what must be done when you multiply or divide an inequality by a negative number?

reverse the inequality sign

F(x-c)

right c units

i

root-1

root

roots refer to the values of x that make f(x)=0

length of line formula

square root all - (x2-x1)^2 + (y2-y1)^2

for inequality questions, what is standard procedure

start with the midpoint of the desired interval, and subtract it from w. make sure that when you subtract it it won't be greater than 1 or whatever

find the roots of v=5t-t^2

t=5, t=0

when a question asks for the solutions to f(x)=k, what is it referring to?

the intersection between f(x) and the line y=k

what is the vertex

the midpoint of the parabula

the number of solutions of a system of equations is the same as what?

the number of intersections between them on the graph

relationship between the number of solutions and the number of intersection points

the number of solutions is equal to the number of intersection points

how do you know if two lines are perpendicular

the products of their slopes is -1.

when a quadratic has just one root, the x-coordinate of the vertex is?

the same as the root

Range for a function

the set of all possible output values from a function

how to calculate the x-coordinate of the vertex?

the x-coordinate of the vertex is always the midpoint of the 2 roots, which can be found by averaging them.

discriminant < 0

there are no real roots

discriminant > 0

there are two real roots (two solutions)

discriminant=0

there is one real root

if x=5 makes a function undefined, how will this appear on a graph?

there will be a vertical asymptote with equation x=5 (ie no y-value when x=5)

what do you do when two lines have infinitely many solutions?

they are either the same equation or one is a multiple of another

how do you know if two lines are parallel .

they have the same slope

semiannually

twice per year

f(x) + c

up c units

ratio of area to side lengths in similar triangles

when 2 triangles are similar, the ratio of their areas is equal to the square of the ratio of their sides. If the ratio of their sides is 3:5, the ratio of their areas is 9:25

absolute value graph

will never reach a negative point as the absolute value of something can never be negative

quadratic equation

x = [-b ± √(b² - 4ac)]/2a

what to remember when looking at graphs?

x and y increments of units may not be the same

REMEMBER WHEN PLUGGING INTO EQUATION

x can be a fraction or negative

what is the domain of y=rootx

x is equal to or greater than 0, because we can't take the square root of a negative number

weird equation like (x+a)^2 = x^2 + 8x +b, what is the value of b?

x^2 + 2ax +a^2 = x^2 +8x+b. MATCH THE COEFFICIENTS. 2a=8 and a^2 = b.

xy=8, xz=5, yz=10. What is the value of xyz?

x^2y^2z^2=400. Square root for xyz=20.

point-slope form

y-y1=m(x-x1)

vertex form

y=a(x-h)^2 + k