Precalculus Final Review

Front 1

Distance Formula

Back 1

▫︎▫︎▫︎▫︎▫︎▫︎_________________

d = √ (x₂- x₁)² + (y₂- y₁)²

Front 2

Midpoint Formula

Back 2

⎧ x₁+ x₂ ▫︎▫︎ y₁+ y₂ ⎫

⎪--------- , ---------⎜

⎩ ▫︎▫︎▫︎2▫︎▫︎▫︎▫︎▫︎▫︎ 2▫︎▫︎▫︎▫︎▫︎⎭

Front 3

How to solve h(x) = g(x) by the Intersection Method

Back 3

1. Graph y₁ = h(x) and y₂ = g(x).

2. Find each point of intersection.

3. The solutions are the x-values of each point of intersection.

Front 4

How to solve h(x) = g(x) by the Zero/X-Intercept Method

Back 4

1. Rewrite the equation as h(x) - g(x) = 0.

2. Graph y₁ = h(x) - g(x).

3. Find the zeros/x-intercepts.

4. The solutions are the x-values of each zero.

Front 5

Standard form of a line

Back 5

𝐴𝑥 + 𝐵𝑦 = 𝐶

Front 6

Slope of a line in standard form

Back 6

m = −𝐴 ⁄𝐵

Front 7

Slope-Intercept form of a line

Back 7

y = mx + b

Front 8

Point-Slope form of a line

Back 8

y - y₁ = m( x - x₁)

Front 9

Relation

Back 9

A set of ordered pairs

Front 10

Domain

Back 10

Set of all input/x-values of a relation

Front 11

Range

Back 11

Set of all output/y-values of a relation

Front 12

Function

Back 12

A special relation such that each input corresponds to exactly one output.

Front 13

Independent variable

Back 13

x

Front 14

Dependent variable

Back 14

y

Front 15

If a function contains an even radical, then its domain is evaluated by

Back 15

setting the "inside" of the radical ≧ 0 .

Front 16

If a function has a fraction in it (or is a fraction itself), then its domain is evaluated by

Back 16

setting the denominator ≠ 0 .

Front 17

Difference Quotient

Back 17

Front 18

A function is even if for every value of x,

Back 18

f ( -x ) = f ( x )

Front 19

An even function is symmetric with respect to the

Back 19

y - axis

Front 20

A function is odd if for every value of x,

Back 20

f ( -x ) = - f ( x )

Front 21

Informally, a function is continuous if

Back 21

its graph can be drawn without lifting your pencil from the paper.

Front 22

Vertical shift (with parent function f(x) )

Back 22

g( x ) = f( x ) ± c

Front 23

Horizontal shift (with parent function f(x) )

Back 23

g( x ) = f( x ± c)

Front 24

Reflection over x-axis (with parent function f(x) )

Back 24

g( x ) = - f( x )

Front 25

Reflection over y-axis (with parent function f(x) )

Back 25

g( x ) = f( -x )

Front 26

Vertical stretch/compression (with parent function f(x) )

Back 26

g( x ) = c × f( x)

c > 1 stretch

0 < c < 1 compression

Front 27

Horizontal stretch/compression (with parent function f(x))

Back 27

g( x ) = f( c × x )

0 < c < 1 stretch

c > 1 compression

Front 28

Two things you must do when solving a radical equation are

Back 28

1. isolate the radical

2. check your answers

Front 29

Discriminant of a quadratic

Back 29

b² - 4ac

> 0 : 2 real solutions

= 0 : 1 real solution

< 0 : no real solutions

Front 30

The x-value (h) of the vertex (h,k) of a quadratic (parabola) in polynomial form (ax² + bx + c) can be found by

Back 30

▫︎▫︎▫︎▫︎▫︎ - b

h = ------

▫︎▫︎▫︎▫︎▫︎ 2a

Front 31

Definition of absolute value

Back 31

▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎⎧x if x ≧ 0

|x| =

▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ ⎩-x if x < 0

Front 32

Properties of polynomial functions

Back 32

1. Domain: ( -∞, ∞ ) or ℝ

2. Continuous

3. Graphs are smooth & rounded

Front 33

End behavior of an odd function

Back 33

Ends go in opposite directions

a > 0 down on right, up on left

a < 0 up on right, down on left

Front 34

End behavior of an even function

Back 34

Ends go in same direction

a > 0 up

a < 0 down

Front 35

A polynomial function of degree n has at most ___ zeros.

Back 35

n

Front 36

A polynomial function of degree n has at most ___ turning points/local extrema/local max/min.

Back 36

n - 1

Front 37

A polynomial function of degree n has at most ___ points of inflection (a point of inflection is a change in concavity).

Back 37

n - 2

Front 38

A product statement (aka remainder polynomial) is of the form

Back 38

f( x ) = d( x ) × q( x ) + r( x )

d( x ) : divisor q( x ) : quotient r(x ) : remainder

Front 39

Remainder Theorem

Back 39

If 𝒇 is a polynomial and 𝒇 is divided by (x - a), then the remainder is f( a ).

Front 40

Factor Theorem

Back 40

If 𝒇 is a polynomial function, then (x - a) is a factor of 𝒇 ⇔ f( a ) = 0.

Front 41

You can find the output of a function 𝒇 at x = c using the remainder r by doing as follows:

Back 41

r = f( c )

Front 42

The possible rational roots of a polynomial function with integral coefficients can be found by

Back 42

factor of constant

------------------------------

factor of leading coef.

aka p/q

Front 43

In general, given a complex number a + bi, then there exists another complex number

Back 43

a - bi, called the conjugate.

Front 44

Patterns of i

Back 44

i = i

i² = -1

i³ = -i

i⁴ = 1

Front 45

Absolute value/ modulus

of a complex number a + bi

Back 45

▫︎▫︎________

√ a² + b²

Front 46

When graphing a complex number,

Back 46

the vertical axis is the imaginary axis

and

the horizontal axis is the real axis

Front 47

Rational function

Back 47

of the form

g( x ) / h( x )

where g( x ) and h( x ) are polynomial functions and h( x ) ≠ 0

Front 48

Vertical asymptote of a rational function

Back 48

set denominator = 0 and solve for x

Front 49

Horizontal asymptote of a rational function

where m is the degree of the numerator

and n is the degree of the denominator

Back 49

m = n → HA: y = a/p

m < n → HA: y = 0

m > n → NO HA

Front 50

Slant asymptote

Back 50

y = q( x )

where q( x ) is the quotient when the numerator of f( x ) is divided by the denominator

Front 51

Holes of rational functions

Back 51

If one or more factors of the function cancel, use the resembles equation to find ordered pair/s for any hole/s

for example: f(x)= [(x+a)(x+b)] / (x+b) → the resembles EQ is g(x) = x+a, and the ordered pair for the hole is (b, g(b))

Front 52

Limits of rational functions

Back 52

Divide the numerator and denominator by x raised to the highest power in the denominator, like so:

Front 53

How to solve polynomial inequalities with a degree > 2

Back 53

1. Write the polynomial so the leading coef. is positive and on the left hand side, and so the inequality is set to 0.

2. Factor the polynomial.

3. Identify critical numbers and region test.

note: same general steps can also be used for rational inequalities

Front 54

A function is one-to-one ⇔

Back 54

its inverse is also a function.

This means each input has exactly one output and each output came from exactly one input. In other words, if f(a) = f(b), then a = b.

Graphically, it must pass both the vertical line test and the horizontal line test. m

Front 55

aᵐ × aⁿ =

Back 55

aᵐ⁺ⁿ

Front 56

aᵐ/aⁿ =

Back 56

aᵐ⁻ⁿ

Front 57

a⁻ᵐ =

Back 57

1/aᵐ

Front 58

aᵐ/ⁿ =

Back 58

▫︎▫︎▫︎▫︎▫︎______

ⁿ√aᵐ

Front 59

x = aʸ ⇔

Back 59

y = logₐx

Front 60

Properties of logarithms

Back 60

Front 61

Change of base formula

Back 61

▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ logv

logₐv = --------

▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎▫︎ loga

Front 62

Compound interest formulas

Back 62

Set time: A = P(1+r/n)ⁿᵗ

Continuously: A = Peʳᵗ

Where P = principal/starting amount, r = rate (percent in decimal form),

t = time (in years), and n = number of compounds (yearly)

Front 63

Exponential law of growth or decay

Back 63

A(t) = A₀eᵏᵗ

Where A₀ is the initial amount when t (time) = 0

​and k is a constant that represents the rate of growth or decay

Growth: k > 0

Decay: k < 0

Front 64

For an arc length s, with radius r and angle θ (in radians), the arc length s can be found with the formula

Back 64

s = rθ

Front 65

For an arc length s, with radius r and angle θ (in radians), the area 𝑨 of the arc can be found using the formula

Back 65

𝑨 = ½θr²

Front 66

The three Pythagorean Identities (of trigonometry) of an angle, θ, are

Back 66

cos²θ + sin²θ = 1

tan²θ + 1 = sec²θ

cot²θ + 1 = csc²θ

Front 67

Sine Half-Angle Identity

Back 67

Front 68

Cosine Half-Angle Identity

Back 68

Front 69

Tangent Half-Angle Identity

Back 69

Front 70

Sine Double Angle Identity

Back 70

sin2θ = 2sinθcosθ

Front 71

Cosine Double Angle Identities

Back 71

cos2θ = cos²θ - sin²θ

cos2θ = 1 - 2sin²θ

cos2θ = 2cos²θ - 1

Front 72

Tangent Double Angle Identity

Back 72

Front 73

Sum & Difference Angle Identities

Back 73

Front 74

Heron's Formula for the area of a triangle

Back 74

Front 75

Sine formulas for the area of a triangle

Back 75

Front 76

Law of Cosines

Back 76

Front 77

To convert rectangular coordinates (x,y) into polar coordinates (r,θ) , let r = ?

Back 77

▫▫▫▫▫▫_______

r = √ x² + y²

Front 78

To convert rectangular coordinates (x,y) into polar coordinates (r,θ) , let θ = ?

Back 78

θ = tan⁻¹ (y ⁄ x)

Front 79

How to multiply & divide complex numbers in polar form

Back 79