Precalculus

Factoring with Powers

Use difference of cubes formulas (ADDITION AND SUBTRACTION) and difference of squares

If there are fractions that you need to factor, take out the COMMON DENOMINATOR

If whole expressions have an exponent, turn them into VARIABLES

Simplifying Rational and Radical Expressions

You might need to EXPAND certain expressions to simplify them

If you want to rationalize the denominator by multiplying it by both sides, make sure it’s REDUCED first so you don’t veer off course

To rationalize a denominator that’s an expression, you multiply by the CONJUGATE

If it asks you to rationalize, do that even if it looks simplified

Sometimes you may just have to leave the expressions multiplied without writing down the result

To cancel expressions you might need to factor out a NEGATIVE

Solving equations

Check ALL solutions to make sure none are extraneous

Isolate radical expressions to square easier

REMEMBER that even roots is plus or minus

You MIGHT need to do integer zero theorem for conventionally unfactorable expressions

To express an absolute value expression as a piecewise function, make it equal to zero and solve. The positive one will have x >= and the negative (applying negative 1) x<

When solving with one absolute value expression make 2 cases where one is negative and the other positive.

With more than one absolute value expression, make them equal to zero, solve, and make a number line with the solutions. Make inequalities and guess and check to see if it makes the radical expressions positive or negative. Then make cases to see if those solutions follow the inequality.

BE CAREFUL. Just because 2 inequalities make the expression positive, doesn’t mean the solution matches the INEQUALITY.

Inequalities

The union sign (upright U) represents “or”. It basically includes all numbers from both sets and rejects none of then

The intersection sign (downturned U) reps “and”, which only included the overlap between both sets.

Solve 3 part inequalities SEPARATELY and denote that separation with an intersection sign. BUT put the overlap tg in ur final answer.

x²>= -a: Any real number works here b/c it will be 0 or greater anyway

x²<= -a: Any real number doesn’t work here b/c it will be 0 or greater and those aren’t less than negatives

When doing rational/quadratic inequalities, put everything to the side and factor, state critical values ( i.e f(x)= 0 when x=… or f(x)=no solution when x=npv).

Remember that denominators and npvs are included in sign analysis. In fractions, only the NUMERATOR can make things equal to zero.

Denote your solution with the union sign when more than one sub domain works

If there is no underline under the inequality sign then all critical values have open circles/round brackets

|x|= a if & only if x=+-a

|x|<a if & only if -a<x<a

|x|>a if & only if x>a “or” x< -a denoted with a union sign

With a case analysis of 2 absolute values on each side of the inequality sign, you have to consider four cases where both r negative/positive, or one of the other are positive and negative. You also have to consider the inequalities that make those cases, to see if the contradict each other.

Once you’ve solved the inequalities make sure to overlap them.

For normal inequalities, put everything to 1 side and make sure if the result is positive or negative. (ex if it’s negative then the num and denom will be either ±). You have to make 2 cases to reflect this (making the num less than zero for example to rep -). You then guess and check each inequality to see what works

Coordinate Geometry

Point-Slope form: y-y_1= m(x-x_1)

If you are given 2 coordinates any work

General form is Ax + By + C= 0 (ALL INTEGERS NO FRACTIONS)

Parallel lines have the same slope and perpendicular lines have negative reciprocal slopes

Sketching functions using transformations

First factor out the first coefficient from the first and second term. Then do (b/2)² and add the positive and the negative versions. Multiply the factored out coefficient by the negative and add it to constant while keeping it at the front.

The maximum value is the y value

Absolute values cannot be negative

Domain and Range

Inside of the square root has to be more or equal to zero so solve that way

For finding the domain you might need to use a number line to see if every element works for fractions w/o radical denominators.

If you have smth like x³ + 1 as the denominator just solve for zero like that

Properties of Functions

For radicals inside radicals do the inner one and then do the outer one with the inner one involved to get the domain.

ALWAYS STATE RESTRICTIONS WITH ANSWERS

RESTRICTIONS FOR RADICALS STILL PERSIST EVEN WHEN IN NUMERATOR

Common misconceptions

Don’t bother putting the intersection symbol and just put the overlap set notation

SIMPLIFY MIGHT MEAN EXPAND

BRACKETS FOR SUBTRACTION

LEAVE EXPRESSIONS FIRST TO SEE IF U CAN SIMPLIFY

SQUARE ROOT OF EXPRESSION W X IS +-

When going from point form to general form get rid of the fraction FIRST

APPLY THE X EVERYWHERE; WHEN YOU’RE DOING COMPOSITE FUNCTIONS THE ORIGINAL NPV OF THE OUTSIDE DOESNT MATTER IF ITS SMTH LIKE 6-x