Precalculus

Rational Number

-any number that is a real number that can be written as a terminating or repeating decimal

Irrational Number

-real number that cannot be written as a terminating or repeating decimal (square root of 5)

Quadratic Formula

-b±[√b²-4ac]/2a

Factoring Cubes

a^3+b^3=(a+b) (a^2-ab+b^2)

-if it is A^3-B^3 then change to (a-b)(a^2+ab+b^2)

Discriminant

-for quadratic expression, the discriminant is b^2-4ac

-determines tht nature of the roots when the quadratic equation=0

-if discriminant is greater than 0, 2 real roots

-if discriminant=0, 1 real root

-if discriminant is less than 0, 2 imaginary roots

Discriminant with x intercepts

-greater than 0, 2 x intercepts

-equal to 0, the vertex lies on the x axis

-less than 0, the parabola doesn't intersect the x axis

Given roots to quadratic equation and have to find the original equation

-formula is

X^2-(sum of roots)x+(product of roots)=0

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Midpoint Formula

(x₁+x₂)/2, (y₁+y₂)/2

-is coordinate set

Circle with center at origin

X^2+Y^2=R^2

Circle with center at (h,k)

(X-h)^2 +(y-k)^2=r^2

-if a line is tangent to a circle, the line is perpendicular to the radius drawn to the point of tangency

Slope intercept form

y=mx+b

-m=slope

-b=y intercept

point slope form

y-y1=m(x-x1)

-passes through point (x1,y1)

standard form

ax+by=c

Parallel Lines

-have equal slopes

Perpendicular Lines

-have slopes that are opposite and reciprocal

Constant of proportionality with K

look at other slides

Y=Kx

-y varies as x

-y varies directly as x

-y is directly proportional to x

Y=K/x

-y is inversely proportional to x

Y=kxz

-y varies jointly as x and z

Y=Kx/z

-y varies directly as x and inversely as z

Y=Kx+b

-y varies linearly as x

relation

- a set of ordered pairs

function

- a relation in which each input has a unique output

domain

-the set of input values

range

-the set of corresponding output values

zeros

-the input values that make the output zero

-values for which f(x)=0

-located on the x axis

Vertical Line test

a relation is a function and passes the vertical line test if when a vertical line is drawn through the graph, it touches the graph only ONE time.

Constant

f(x)=a

Linear

f(x)=ax+b

Quadratic

f(x)=ax^2+bx+c

Cubic

f(x)=ax^3+bx^2+cx+d

Quartic

f(x)=ax^4+bx^3+cx^2+dx+e

Triangle

A=(B)(H)/2

- formula is one half base times hight

Rectangle

A=bh

Parallelogram

A=bh

Trapezoid

A=(1/2)(B+b)(H)

Circle

A=@r^2

@=pie

C=2(pie)r or (pie)(diameter)

Prisms

V=(area of base)(height)

Pryamids

V=(1/3)(area of base)(height)

Cube

V=(edge)^3

S.A.= 6(edge)^2

Rectangular Prism

V=(lenght)(width)(height)

S.A.=2(lenght)(width)+2(lenght)(height)+2(width)(height)

Right circular cylinder

V=(pie)r^2h

S.A.=2(pie)r^2+2(pie)RH

Sphere

V=(4/3)(pie)r^3

S.A.=4(pie)r^2

Right Circular Cone Formula

V=1/3πr^2h

SA=πr√(r^2+h^2 )+πr^2

Distance

D=(rate)(time)

Position Function

-16t^2+v0t+h0

V0=initial velocity

H0=initial hieght

Finding Domain for a rational function

-omit numbers that make the denominator zero

Finding domain for radical functions

-the radicand must never become negative so find domain by solving radicand> or equal to 0

Difference Quotient

-gives the secant slope to a curve

f(x+h)-f(x)/h

Increasing Function

A function whose output value increases as its input value increases.

-graph is increasing between (a,b) when F(b)>F(a) when b>a

-graph rises to the right

Decreasing

opposite

-graph falls to the right

constant

as move from a to b, f(b)=f(a)

-graph is horizontal

Basic shapes

look at next ones

Y=absolute value of x

Looks like a V that intersects at the origin

y=square root of x

is a line that comes from the x axis that extends rightward. Gets less steep as it moves along

Y=x^2

Is a parabala im pretty sure. Looks like the square root of x one but intersects at the origin and has two sides

Y=x^3

flip the left side of the x^2 side down

Y=1/x

don't even know how to explain

Y={x}

-don't even know

y=square root of A-x^2

semicircle on top half

f(x)+d

-vertical shift with the sign

F(x+c)

horizontal shift against the sign

Af(X)

-vertical stretch or squeeze

-multiply each y value by A

f(Bx)

-horizontal stretch or squeeze

-f(x)

-reflect graph across x axis

f(-x)

-reflect graph across y axis

absolute value of f(x)

-keep all positive parts of graph

-flip negative parts up

f(absolute value of x)

-keep positive x function value

-mirror those values across the y axis

Even functions

f(-x)=f(x)

Odd functions

f(-x)=-f(x)

(f+g)(x)

F(x)+g(x)

(f-g)(x)

F(x)-g(x)

(F*g)(x)

F(x)*G(x)

(f/g)(x)

F(x)/G(x)

-g(x) can't equal zero

Finding inverse of a functions

switch x and y and solve for the new y

Invertable functions

finverse{f(x)}=x

parabola standard form

y=ax^2+bx+c

-a gives shape

=1 is normal, >1 is fat,<1 is skinny(absolute values)

-- if a >0 opens up, <0 opens down

-c is the y intercept

-b gives movement on the horizontal axis

find the vertex by x=-b/2a, y=f(-b/2a)

Completed square form for a parabola

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

a gives shape

(h,k) is the vertex

intercept form for a parabola

y=a(x-r1)(x-r2)

a gives shape

r1 and r2 are the x intercepts

vertex x=(r1+r2)/2 y=f((r1+r2)/2)

discriminant with parabola

b^2-4ac

>0 then the parabola has 2 x intercepts

<0 parabola has 0 x intercepts

=0 the parabola has a vertex on the x axis

Circle formula

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

-center is (h,k)

-radius is r

Ellipses

-set of points(x,y) in a plane, the sum of whose distances from two distinct fixed points(foci) is constant

(x-h)^2/a^2 + (y-k)^2/b^2 =1

-can swap the numerators

-has center (h,k)

-eccentricity=c/a

a^2+b^2=C^2

-c gives the distance to the focus from the center

-eccentricity is less than one

Hyperbolas

-set of all points(x,y) in a plane, the difference of whose distances from two fixed distinct points (foci) is a positive constant

(x-h)^2/a^2 - (y-k)^2/b^2 = 1

-can switch the numerators

-has center (h,k)

c^2=A^2+b^2

-c gives the distance from the focus to the center

-asymptotes pass through (h,k) and have slope (+/-) b/a or (+/-) a/b

-equation is y-k=(+/-)b/a(x-h)

-eccentricity is greater than 1

Parabolas

-set of all points P in a plane such that the distance from P to a fixed point F(the focus) is equal to the distance from P to a fixed line D( the directrix)

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

-has a vertex (h,k)

a=1/4c or c=1/4a

-absolute value of c is the distance from vertex to focus or vertex to directrix

Polynomials

-continous smooth functions having no corners or gaps

-no horizontal segments

-if the degree is n, the graph has at most n x axis

-maximum number of turning points is n-1

-polynomial of degree n has at most n-1 local extrema

-at zero of even multiplicity, the graph touches but doesn't cross the x axis

-at zero of odd multiplicity, the graph crosses the x axis

-if degree is even, the graph ends in the same direction

-if the degree is odd, the graph ends in opposite directions

-if the leading coeffienct is positive, the graph ends in quadrant 1

-if the leading coefficient is negative, the graph ends in quadrant 4

Chapter 4.2 need help

need blakes help

-use synthetic division

ex: x-1 put opposite of c(1) in box. Then take coefficients of the formula and put them up. Pull first one down and then multiply across.

i

square root of negative one

i^2

-1

i^3

-i

i^4

1

rationalizing i in the numerator

1/(a+bi)

-multiply by (a-bi)/(a-bi)

intermediate value theorem

-since polynomial functions are continuous, if f(a) and f(b) are of opposite sign, there is at least one zero of f between a and b

chapter 4.8 need blakes help

need blakes help

Angel measurements

angels can be measured in degrees, minutes, seconds or decimal degrees

-1 degree is equal to 60 minutes (60')

-1 minute is equal to 60 seconds (60'')

Converting between radians and degrees

multiply by (pie)/180 degrees or 180/(pie)

positive and negative angles

-positive angles are measured counterclockwise

-negative angles are measured clockwise

Coterminal angles

-have same initial and terminal sides and look identical

-find by adding or subtracting 360 degrees, 2pie radians or 1 revolution

Sin&

=y/r

Cos&

=x/r