Math Midterm 11/14

Vertex Form Vertex

(h,k)

argument

(of a function) any value substituted for x

Cross-multiplying

*solve and check for extraneous solutions*

Limits of graphs

+: going from right -: going from left - when there is a hole the limit equals the y value from both directions - when there is a VA decide whether the graph is going towards ∞ or -∞ - when there is a HA decide whether the graph is going towards ∞ or -∞

end behavior asymptote

- when there is an HA the EBA is the HA - when there is no HA, use synthetic division to find the EBA (it is the answer without the remainder)--slant asymptotes

i²=

-1

Integers

...-3,-2,-1,0,1,2,3..

Whole Numbers

0,1,2. . .

how to find holes

1) factor the numerator and the denominator 2) find the zero of the common factor 3) rewrite the function without the common factors, then plug in the zero as the x value to find the y value

Table of signs

1) find the x-intercepts and the vertical asymptotes which will determine the intervals 2) use all the factors and a number in each interval to determine the sign of that portion of the graph 3) plot the intercepts and the asymptotes, then sketch the graph +: above x axis -: below x axis

Different ways to factor:

1) pull out common factors 2) difference of two squares 3) Un-FOIL

Natural Numbers

1,2,3. . .

A linear function has ____ constant differences.

1st

A quadratic function has ____ constant differences.

2nd

A cubic function has ____ constant differences.

3rd

Polynomial

A monomial or the sum of monomials

Compound Interest

A=P(1+r/n)∧nt A=amount at any time P=initial amount r=annual interest rate as a decimal n=number of times interest is compounded in a year t=number of years

continously compounded interest

A=Pe^rt A=amount at any time P=initial amount r=annual interest rate as a decimal t=number of years

Real Numbers

All the numbers on the number line (including fractions and decimals) --> both rational and irrational

Radical

An expression that has a root (e.g., square root, cube root). Example: √36 is a radical.

Solving Quadratic Inequalities

Graph the function (this will help) What values of x make the statement true

Complex Numbers

Imaginary numbers and real numbers together make up the set of these numbers

Log is the __________ operation of _____________.

Log is the inverse operation of exponents.

Irrational Numbers

Numbers that can't be written as a fraction... examples √2 and π

Logarithm

The base 10 logarithm of n is a if 10^a = n (which means log n = a). The natural logarithm of n is a if e^a = n (which means ln n = a). The number e (= 2.71828...) is called the base of the natural logarithm.

vertex

The point (x,y) of a parabola where it crosses the axis of symmetry. (highest or lowest point)

Conjugate

When the denominator is a mix of rational and an irrational number, must multiply both numerator and denominator by the same numbers in the same order but a different sign.

Use inverse matrix to solve linear system

[A]-¹ × [B] = [. . .]

Dilation of simple rational functions

a

piecewise function

a function composed of 2 or more functions each with its own domain--open circle is only for less than or greater than

composition of functions

a function is performed, and then a second function is performed on the result of the first function -- f(g(x))

Rational Function

a function of the form f(x)=p(x)/q(x) where p(x) and q(x) are polynomials and q(x) ≠ 0

inverse of a function

a function that "undoes" what the original function does -- switch x and y then solve for y

polynomial function

a function that is represented by a polynomial equation

function

a relation in which each element of the domain is paired with exactly one element of the range

greatest integer function

a step function, written as f(x)=[[x]], where f(x) is the greatest integer less than or equal to x.

Rational Numbers

all numbers that can be written as positive or negative fractions ...rational numbers can be expressed as fractions, decimals, or percents

domain in odd roots

all real numbers

relation

any set of ordered pairs

Zeros of a polynomial

as many as the highest exponent

Double zero

bounces off x axis

Discriminant

b²−4ac

leading coefficient

coefficient of the first term when the polynomial is in standard form

Horizontal asymptote of general rational functions

compare degree of numerator and denominator if °(n)>°(d) NO HA if °(n)<°(d) HA: y=0 if °(n)=°(d) HA: coef.(n)/coef.(d)

Trinomial=

cubic

Maximum turns in a polynomial

degree-1=max amount of turns

Complex zero

doesn't cross through x axis *always comes in pairs*

Direct substitution

evaluating a function by plugging a given value into the function

odd function

f(-x)=-f(x) graph is the same when reflected over origin

rational algebraic function

f(x) = p(x)/q(x) --p and q are polynomial functions

even function

f(x)=f(-x) graph is the same when reflected over y axis

absolute value parent function (+piecewise)

f(x)=|x| = { x, x>=0 -x, x<=0

End behavior: even, negative

f(x)→ -∞ as x→∞ f(x)→ -∞ as x→-∞

End behavior: odd, negative

f(x)→ -∞ as x→∞ f(x)→ ∞ as x→-∞

End behavior: odd, positive

f(x)→ ∞ as x→∞ f(x)→ -∞ as x→-∞

End behavior: even, positive

f(x)→ ∞ as x→∞ f(x)→ ∞ as x→-∞

derivative of ax^n

f/(x)=a(n)xⁿ⁻¹ (slope of tangent)

y=f(|x|)

for x>0 y values remain the same -- those y values are used for the negative of the positive values of x for x<0 y values are eliminated

Graphing Quadratic Inequalities

graph the normal parabola (dotted if < or >) choose a testing point and plug it into the equation if the statements holds true, shade the area the testing point is part of

Vertical asymptote of simple rational functions

h

y=f(x/b)

horizontally dilated --fraction stretches --whole number strinks

y=f(x-h)

horizontally translated by h units

Asymptotes are __________ discontinuities.

infinite

Domain of graph when increasing and decreasing

is it going up or down? what is the domain of each section? - use < and >

Horizontal asymptote of simple rational functions

k

a (subscript) n

leading coefficient

Monomial=

linear

Exponent Property

log₁(x)²=2log₁(x) the ₁ represents b the ₂ represents a

Division Property

log₁(x/y)=log₁(x)-log₁(y) the₁ in this case represents b. . . the property works as long as bases are the same.

Multiplication Property

log₁(xy)=log₁(x)+log₁(y) the ₁ in this case represents b. . . the property works as long as bases are the same.

Change of base

log₁b=(logb/log1) the ₁ represents the base which can be any number which will correspond to the number in the divisor

b²-4ac<0

no real solutions

Imaginary Numbers

number that can be written as a real number multiplied by the imaginary unit i

b²-4ac=0

one solution

Hyperbola

parent function: f(x)=1/x

Binomial=

quadratic

4th term poly.=

quartic

y=-f(x)

reflection across x-axis

y=f(-x)

reflection across y-axis

y=-f(-x)

reflection over x-axis and y-axis (origin)

Holes are __________ discontinuities.

removeable

Y-Intercepts of general rational functions

set all x values equal to zero, solve

X-Intercepts of general rational functions

set numerator equal to zero, solve

Domain of a log function

set the argument >0

Vertical asymptote of general rational functions

set the denominator equal to 0, solve

quadratic function

the graph changes direction at one vertex (is a parabola) y=ax²+bx+c y=x² (x1,y1) (x2,y2) (x3,y3), write three equations using y=ax²+bx+c, solve using matrix

Degree of a polynomial

the highest exponent OR how many branches in the graph

y=|f(x)|

the positive y values remain the same and the negative y values become positive (entire graph goes into the first two quadrants)

domain of a rational function/inverse variation

to find the domain, set the denominator equal to zero and solve for the variable (what will make it undefined?) -- gets vertical asymptote

domain in square root and other even roots

to find the domain, set the radicand (under the root) to be greater than or equal to zero and solve

b²-4ac>0

two real solutioons

How to write a quadratic equation

use what numbers you have to make part of the equation plug a point in (x,y) and solve for the last remaining variable you need

y=a*f(x)

vertically dilated

y=f(x)+k

vertically translated by k units

Invertible function

when a graph and it's inverse are both functions (aka one-to-one)

Standard Form Vertex and Y-Intercept

x=-b/2a plug in x and solve for y y intercept is c

Vertex Form

y = a(x-h)² + k

direct variation function

y = ax --straight line passes through the origin --domain is x>=0

Simple rational functions

y=(a/(x-h))+k

General rational functions

y=(ax+b)/(cx+d)

exponential function

y=a(b)^x y=b^x set up a system of equation, substitute, solve has y intercept, doesn't go through the origin

Intercept Form

y=a(x-r₁)(x-r₂)

inverse variation function

y=a/x --both of the axis are asymptotes --doesn't go through origin --used for real life problems (no negative answers)

power function

y=ax^b y=x^b set up a system of equation, substitute, solve using logs crosses origin

Standard Form

y=ax²+bx+c

linear function

y=mx+b y=x slope: change in y over change in x, use slope, solve for y intercept a line

Intercept Form X-Intercepts and Vertex

zeros: r₁ and r₂ find the average of the zeros which is the x value of the vertex plug that value in as x to get y