Raising a negative number to a power

When raising a negative number to a power always put the negative number in a parenthesis.

Factoring ax²+bx+c

Factoring by grouping

Canceling out opposites

If both terms are opposite signs they can be crossed out and replaced by a negative 1.

Factoring out a -1

To factor out a -1, remove the negative and change the signs of the terms

Operations with fractions

Complex fractions

Equations with fractions

To solve equations with fraction, find a common denominator.

Drop the denominators

Solve the resulting equation.

Check the answer(s) in the denominators for extraneous roots.

Drop the denominators

Solve the resulting equation.

Check the answer(s) in the denominators for extraneous roots.

Quadratic formula (Find the Roots)

the quadratic formula is x equals negative b plus or minus the square root of b squared minus 4 times a times c all over 2 times a.

Completing the square

Discriminant (describe the roots)

if the discriminant is 0, the roots are real, rational and equal.

if the discriminant is negative, the roots are complex or imaginary.

if the discriminant is a perfect square, the roots are real, rational, and unequal.

if the discriminant is not a perfect square, the roots are real, irrational, and unequal

if the discriminant is negative, the roots are complex or imaginary.

if the discriminant is a perfect square, the roots are real, rational, and unequal.

if the discriminant is not a perfect square, the roots are real, irrational, and unequal

Sum and product of roots

The sum of the roots is negative b divided by a.

The product of the roots is c divided by a.

The equation for sum and product of the roots is x squared minus the sum times x plus the product equals 0.

The product of the roots is c divided by a.

The equation for sum and product of the roots is x squared minus the sum times x plus the product equals 0.

Absolute value inequalities

To solve absolute value inequalities

Isolate the absolute value on one side of the equation.

Change the inequality sign and the sign(s) of the terms on the right to the opposite signs.

If the original inequality has a greater than or greater than or equal to (≥ or >) sign use or between your solutions.

Solution is <(≤) x or x >(≥) solution.

If the original inequality has a less than or less than or equal to (≤ or <) write as an interval. Solution <(≤) x <(≤) solution. You have to divide by a negative, do not forget to flip the inequality sign.

Isolate the absolute value on one side of the equation.

Change the inequality sign and the sign(s) of the terms on the right to the opposite signs.

If the original inequality has a greater than or greater than or equal to (≥ or >) sign use or between your solutions.

Solution is <(≤) x or x >(≥) solution.

If the original inequality has a less than or less than or equal to (≤ or <) write as an interval. Solution <(≤) x <(≤) solution. You have to divide by a negative, do not forget to flip the inequality sign.

Absolute value equations

To solve an absolute value equation

Isolate the absolute value on one side of the equation.

Change the sign(s) of the terms on the right to the opposite sign(s).

check your answer(s) for extraneous roots.

Isolate the absolute value on one side of the equation.

Change the sign(s) of the terms on the right to the opposite sign(s).

check your answer(s) for extraneous roots.

Inverse variation

The inverse variation formula is x one times x two equals y one times y two.

Inverse variation creates a corner hyperbola.

Inverse variation creates a corner hyperbola.

Conjugate

To find the conjugate

The first term stays the same.

Change the sign of the second term.

The first term stays the same.

Change the sign of the second term.

Domain, range, inverse

The domain is the x coordinates.

The range is the y coordinates.

To find the inverse switch the x and y coordinates.

The range is the y coordinates.

To find the inverse switch the x and y coordinates.

Function

To be a function

It must pass the vertical line test.

No x coordinate can repeat.

An equation must have a y, but the y cannot be squared.

It must pass the vertical line test.

No x coordinate can repeat.

An equation must have a y, but the y cannot be squared.

One to one function

A one to one function must pass the vertical and horizontal line test.

No x or y coordinate can repeat.

No x or y coordinate can repeat.

Logarithms

Convert degrees to radians

To convert from degrees to radians multiply the degrees by pi and divide by 180.

Convert radians to degrees

To convert from radians to degrees, multiply the radians by 180 and divide by pi.

Degrees and radians

Once around a circle is 2 pi radians or 360 degrees.

½ of the way around a circle is pi radians or 180 degrees.

¼ of the way around a circle is pi divided by 2 or 90 degrees.

½ of the way around a circle is pi radians or 180 degrees.

¼ of the way around a circle is pi divided by 2 or 90 degrees.

Arc length – radius formula

In a circle the central angle in radians equals the arc length divided by radius.

Circle formula

The circle formula is x minus h squared plus y minus k squared equals the radius squared.

Secant

Secant equals 1 divided by cosine.

Cosecant

Cosecant equals 1 divided by sine.

Cotangent

cotangent equals 1 divided by tangent or

cosine divided by sine.

cosine divided by sine.

Tangent

Tangent equals sine divided by cosine.

Trigonometric identity

Cosine squared plus sine squared equals 1.

Exact values of trigonometric functions

Arithmetic sequence

An arithmetic sequence is created by adding or subtracting a common difference.

Geometric sequence

A geometric sequence is created by multiplying or dividing a common ratio.

If the terms alternate between positive and negative, the common ratio is negative.

If the terms alternate between positive and negative, the common ratio is negative.

Recursive sequences

a to the n minus 1 – look to the term behind.

a to the n plus 1 – loot to the term in front.

a to the n plus 1 – loot to the term in front.

Rationalizing the denominator

To rationalize a denominator multiply the numerator (top) and denominator (bottom)

by the conjugate of the denominator (bottom).

by the conjugate of the denominator (bottom).

Survey, observational study, experiment

Survey – ask

Observation study – watch

Experiment – change

Observation study – watch

Experiment – change

Quadrants

All students take classes.

Sine and cosecant are positive in quadrants 1 and 2.

Cosine and secant are positive in quadrants 1 and 4.

Tangent and cotangent are positive in quadrants 1 and 3.

Sine and cosecant are positive in quadrants 1 and 2.

Cosine and secant are positive in quadrants 1 and 4.

Tangent and cotangent are positive in quadrants 1 and 3.

Sine, Cosine, Tangent on the unit circle

Converting radians to degrees and minutes

Multiply by 180 and then divide by pi.

The part before the decimal is the degrees. Multiply the decimal by 60 to find the minutes.

The part before the decimal is the degrees. Multiply the decimal by 60 to find the minutes.

Conterminal Angles

Coterminal angles are just different ways of naming the same angle.

Arc functions

Area of a triangle (need S.A.S.)

Need side angle side (S.A.S) to find area.

The area of a triangle equals ½ times a times b times sine of C.

The area of a triangle equals ½ times a times b times sine of C.

Area of a parallelogram

A parallelogram is made up of 2 congruent triangles.

Law of Sines

The law of sines is a divided by sine of A equals b divided by sine of B equals c divided by sine of C.

Law of cosines given (S.A.S.)

The law of cosines is b squared plus c squared minus 2 times b times c times the cosine of A.

Take the square root of the answer to find the missing side.

Take the square root of the answer to find the missing side.

Law of cosines given (S.S.S.)

The law of cosines a squared(the side across from the angle) equals b squared plus c squared minus 2 times b times c times the cosine of x (A). Use numerical solve to find the measure of angle A.

Vector problems

To solve vector problems remember that

The opposite sides in a parallelogram are congruent.

The consecutive angles in a parallelogram are supplemental (add up to 180 degrees).

Use law of Cosines to find the resultant and the angle between the 2 forces.

Use law of Sines to find the angle between a force and the resultant.

The opposite sides in a parallelogram are congruent.

The consecutive angles in a parallelogram are supplemental (add up to 180 degrees).

Use law of Cosines to find the resultant and the angle between the 2 forces.

Use law of Sines to find the angle between a force and the resultant.

Ambiguous Case (0,1,2 triangles?

Permutations with repetition

Binomial Expansion

∑ (Sigma)

Square root equations

To solve square root equations

Isolate the square root on one side of the equation.

Square both sides of the equation

Check the answer(s) in the original equation for extraneous roots.

Isolate the square root on one side of the equation.

Square both sides of the equation

Check the answer(s) in the original equation for extraneous roots.

Converting between logarithm and exponent form

To convert from logarithmic form to exponent form the base remains the base and the rest of the stuff crosses.

the b and the n can never be negative.

the b and the n can never be negative.

Common Logarithms

Natural Logarithms

Inverses of exponential and logarithmic equations

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