Basic Algebra, domain and range, HA, VA, etc

How do you get rid of √ ‘s in denominators, especially when isolating a variable?

ex: Get rid of the square root in the expression a / ( (√b) + c )

MULTIPLY BY THE CONJUGATE!

a / ( (√b) + c ) * √b - c / √b - c
↑ Conjugate is there, it is the den with opp signal rep as 1 whole

How do you solve inequalities?

Get everything on one side, get basic equation, find all values that make the eq true by finding all zeros and making a number line

What is domain?

Domain is all real numbers except √(-), den = 0 making it undefined, or anything that makes graph not true/ possible like ln(0) or ln(-)

What is the difference between (ab)² and (a + b)² ?

(ab)² = a² * b²
(a + b)² = (a+b) * (a+b)

What is the transformation equation?

a * f (bx - c)

a = vertical dialation

b = horizontal dialation

c = horizontal shift

d = vertical shift

Note: power of f() changes based on graph being:

x →af (bx - c) or x²→af (bx - c)²

what is the equation of line or point slope form?

y-y₁ = m (x-x₁)

What are the exponential growth/ decay formulas?

Base formula: y = a * b^x

y = final, a = initial, b = constant multiplier, x = time

Note : find b if given dec or % ↓

growth = a (1+r)^x decay = a (1-r)^x
r = rate of growth as a decimal

How to identify horizontal asymtotes?

Num greater power→ no horizontal asymtotes
Den greater power→ y = 0 i.e. HA = 0
Powers are even→ ratio of num greatest power to den greatest power

what is the range of arcsin(x)?

[ - π / 2 , π / 2 ]

what is the range of arccos(x)?

[ 0 , π ]

what is the range of arctan(x)?

[ - π / 2 , π / 2 ]

How do you solve for x intercepts? How does this relate to solving VA’s?

Find x values that make the original eq = 0 but do not make original den = 0. If the x value makes original den = 0 it is undefined unless it can cancel out, which is a hole.

VA→ When den = 0 but cannot be a hole meaning it cannot cancel out

How to factor ax² + bx + c when a = 1?

ex: x² + 5x + 6

a * c and add to b, then insert those values into either:

(x+#) (x+#) or (x-#) (x-#)

3×2 = 6

3+2 = 5
(x+3) (x+2)

How to factor ax³ + bx² + cx + d?

4 terms means try to group first and second half, pull out, and make factors.

How to factor ax² + bx + c when a < 0?
ex: -2x² + 2x + 12

-2 ( x² - x - 6 )
-2(x - 3)(x + 2)

How to factor ax² + bx + c when a > 1?
ex: 2x² + 9x + 9

2 × 9 = 18
6 + 3 = 9
( 2ײ + 6x ) + ( 3x + 9 )
2x ( x + 3 ) + 3 (x + 3 )
(2x+3) (x+3)