Test 1 (2.2-3.5) Flashcards
Slope
m=(y2-y1)/(x2-x1)
Average rate of exchange
Δy/Δx
vertical line
x=a
point slope form of a line
y-y1=m(x-x1)
horizontal line
y=b
slope-intercept form
y=mx+b
general form of a line
Ax+By=C
Standard form of an equation of a circle
(x-h)(squared)+(y-k)(squared)=r(squared)
general form of a circle
x(squared)+y(squared)+ax+by+c=0
direct variation
y=kx
inverse variation
y=k/x
joint variation
y=k(at)/x
the number k is called the
constant of proportionality
when the value of one variable is related to the value of a second variable it is called a
relation
a relation is
a correspondence between two sets
when speaking of functions we say x ___ to y or y ___ on x
corresponds, relies
how do you write x corresponds to y?
x—>y
x is the ___ and y is the ___
input, output
the set of all inputs for a relation is called
the domain
the set of all outputs for a relation is called
the range
a ___ is a special type of relation
function
Definition of a function
a function form X into Y is a relation that associates with each element of X exactly one element of Y
the set X is called the
domain
the set Y is called the
range
the corresponding y to set x is called the
value or the image of x
for a function no input has…
more than one output
for a function, the variable x is called the
independent variable
for a function, the variable y is called the
dependent variable
the independent variable is also called the
argument of the function
the difference quotient of a function f(x)
f(x+h)-f(x)/h when h≠0
function implicit form
x+y=a
function explicit form
y=f(x)=x
steps for finding the domain of a function defined by an equation
- start with the domain as a set of all real numbers
- if the equation has a denominator, exclude any numbers that give a zero denominator
- if the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative.
the sum f+g function is defined by
(f+g)(x)=f(x)=g(x)
the difference f-g function is defined by
(f-g)(x)=f(x)-g(x)
the product f•g function is defined by
(f•g)(x)=f(x)•g(x)
the quotient f/g function is defined by
(f/g)(x)=f(x)/g(x) when g(x)≠0
if any vertical line intersects a graph at more than one point, the graph is
not the graph of a function
what is the vertical line test
a set of points in the xy plane is the graph of a function only if every vertical line intersects the graph in at most one point
a function is ___ if for every number x in its domain the number -x is also in the domain and
even, f(-x)=f(x)
a function is __ if for every number in its domain the number -x is also in the domain and
odd, f(-x)=-f(x)
a function is even only if its graph is symmetrical in respect to__
the y axis
a function is odd only if its graph is symmetrical in respect to__
the origin
when describing the behavior of a function in terms of its increasing, decreasing, or constant, do so in terms of its __ values
x
the __ value is the local minimum or maximum value of a function
y
the absolute minimum and maximum values are also called the
absolute extrema or extrema values
if f is a continuous function whose domain is a closed interval [a,b] then f has an absolute maximum and minimum value of
[a,b]
equation for average rate of exchange of a function
Δy/Δx=f(b)-f(a)/b-a when a≠0
secant line slope equation
f(b)-f(a)/b-a=f(a+h)-f(a)/h
the average rate of change from points a to b equals the slope of the secant line containing the two points
(a,f(a)) and (bf(b))
properties of function f(x)=√x
- The domain and range re a set of nonnegative real numbers.
- the x and y intercept of the graph is 0
- the function is neither even nor odd
- the function is increasing on interval [0,∞]
- the function has an absolute minimum of 0 at x=0
properties of f(x)=∛x(cube root function)
- the domain and the range are a set of all real numbers
- the x and y intercept are 0
- the function is odd
- the function is increasing on interval (-∞,∞)
- the function does not have any local minima or maxima
properties of f(x)=lxl(absolute value)
- the domain is a set of all real numbers, the range is {fly≥0}
- the x and y intercepts are 0
- the function is even
- the function is decreasing on interval (-∞,0] and increasing on interval [0,∞)
- the function has an absolute minimum of 0 at x=0
constant function equation and appearance
f(x)=b when b is a real number, the graph should look like a horizontal line
domain and range of constant function
domain: set of all real numbers
range: set consisting of a single number b
identity function equation and appearance
f(x)=x, graph should appear as a linear line
domain and range of identity function, slope, y intercept
domain and range are a set of all real numbers
slope is 1 and y intercept is 0
this function is odd
domain and range of square function, intercept
domain: set of all real numbers
range: nonnegative real numbers
intercept at 0,0
square function equation and appearance
f(x)=x^2, graph should appear as a parabola
cube function equation
f(x)=x^3
domain and range of cube function and intercept
domain and range are a set of all real numbers
intercept is 0,0
this function is odd
square root function equation
f(x)=√x
square root function domain and range and intercept
domain and range are a set of nonnegative real numbers and the intercept is 0,0, the is function is neither odd not even
reciprocal function equation
f(x)=1/x
domain and range of reciprocal function and intercepts
domain and range are set of all nonzero real numbers, the graph has no intercepts and is an odd function
what does int(x) stand for
largest integer less than or equal to x
equation and properties of int(x)
f(x)=intx=greater integer less than or equal to x
the domain is a set of all real numbers, the range is a set of integers
a function is continuous if
you can draw it without picking up your penciled there are no gaping holes
int(x) is also called
a step function
