Math 114 Quiz 1 questions and answers

a^m x a^n

a^m+n

(a^m)^n

a^mn

a^n/a^m

a^n-m

-1^n if n is even

1

-1^n if n is odd

-1

n√ab

n√a x n√b

n√a if n is odd

a

n√a if n is even

IaI

quadrant I

+x, +y

quadrant II

-x, +y

quadrant III

-x, -y

quadrant IV

+x, -y

distance between two points

d = √(x₁-x₂)² + (y₁-y₂)²

midpoint of two points

(x1+x2/2, y1+y2/2)

slope of line between two points

y2-y1/x2-x1

positive slope

negative slope

zero slope

no slope

slope intercept form

y=mx+b

point slope form

y-y1=m(x-x1)

two lines are parallel if

their slopes are equal

Two lines are perpendicular if

their slopes are opposite reciprocals

x intercept is where the graph intercepts

x axis, set y = 0

y intercept is where graph intercepts

y axis, set x = 0

symmetric about x axis

Equation is unchanged when x is replaced by -x

symmetric about y axis

Equation is unchanged when y is replaced by -y

symmetric about origin

f(-x)=-f(x)

three ways to represent a range of real numbers

inequalities, intervals, number lines

Ix-aI < c

has solutions centered at a=x, and extends c unites from a on either side

quadratic equation

ax² + bx + c = 0

quadratic formula

x = -b ± √(b² - 4ac)/2a

discriminants

b^2-4ac

discriminant is greater than zero

two real solutions

discriminant equals zero

one real solution

discriminant is less than zero

no real roots

extraneous solutions

results that are not solutions to the original equation

if c > 0 then IxI becomes

x=+-c

if A is less than or equal to B then

-a is greater than or equal to 1/b

if a, b > 0 and a < or equal to b then

1/a > or equal to 1/b

if c > 0 then IxI < or equal to c implies

-c < or equal to x < or equal to c

IxI > c implies

x < -c or x > c

if c > 0, Ix-aI < or equal to c has the interval of

solutions centered at a and extending c units, [a-c, a+c]

(+)(+)

positive

(+)(-)

negative

(-)(-)

positive

steps of solving polynomial inequaliaties

1. move all nonzero terms to one side

2. factor, list key numbers on the number line

3. make a table of each factors sign

4. determine the sign of entire expression, check key numbers

true or false, inputs of a function can only have one output

true

domain of a function

all possible x-values, inputs

when determining the domain of a function

a function cannot divide by zero or take the square root of a negative number

a piecewise function

function defined by different formulas for different domains

greatest integer function

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

range of functions

set of all outputs

when finding range of a function

must get all outputs which can occur or get all outputs that cannot occur

vertical line test

a curve is a function if every vertical line intersects it at most once

increasing function

Graph that rises from left to right

decreasing function

Graph that falls from left to right

a function f is ________ on (a,b) if f(x1) < f(x2) for any x1< x2 in (a,b)

increasing

a function f is ____________ on (a,b) if f(x1) > f(x2) for any x1

decreasing

maximum value of a function

The y-value of the highest point on the graph of a function

minimum value of a function

The y-value of the lowest point on the graph of a function

vertical asymptote

a vertical line that a graph approaches but never crosses