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68 terms

x-axis

horizontal real number line

y-axis

vertical real number line

origin

point of intersection of vertical and horizontal axis

quadrants

four regions of coordinate plane

x-coordinate

direct distance from the y-axis

y-coordinate

direct distance from the x-axis

Cartesian plane

an ordered pair of real numbers can be represented in a plane called the rectangular coordinate system

distance formula

a result derived from the Pythagorean Theorem

midpoint formula

finding the average values of the representative coordinates of the two endpoints of a line segment in a Cartesian plane.

solution

an ordered pair that solves an equation when x and y are substituted and the equation is proven true

graph

the set of all solution points of an equation

intercepts

the points at which a graph intersects or touches an axis

y-axis symmetry

when (x,y) is on the graph, (-x,y) is also on the graph

circle; (h,k); r

the equation (x-h)2+(y-k)2=r2 is the standard form of the equation of a ------ with center ---- and radius ---

numerical approach

when you construct and use a table to solve a problem

linear equation

the simplest mathematical model for relation two variable, y=mx+b

slope

for a line, the ratio of the change in y to the change in x

parallel

if & only if slopes are equal

perpendicular

if & only if slopes are negative reciprocals of each other

rate

when the x-axis and the y-axis have different unites of measure, slope can be interpreted as --------.

linear extrapolation

the prediction method used to estimate a point on a line that does not lie between the given points

domain; range; function

a relation that assigns to each element x from a set of inputs, or ------, exactly one element y in a set of outputs, or -------, is called a ------.

verbally; numerically; graphically; algebraically

functions are commonly represented in four different ways

independent variables

for an equation that represents y as a function of x, the set of all values taken on by the ------- variable x is the domain

dependent variables

for an equation that represents y as a function of x, the set of all values taken on by the ----------- variable y is the range.

piecewise-defined function

f(x)= 2x-1 x<0

x2+4 x>=0

x2+4 x>=0

implied domain

if the domain of the function f is not given, then the set of values of the independent variable for which the expression is defined

difference quotient

in calculus, one of the basic definitions is given by [f(x+h)-f(x)]/h h cannot equal 0

ordered pairs

the graph of a function is the collection of ------- or (x, f(x)) such that x is the domain of f

Vertical Line Test

used to determine whether the graph of an equation is a function of y in terms of x

zeros

in a function the values of x for which f(x)=0

decreasing

a function f is ------- on an interval if, for any x1 and x2 in the interval, x1 < x2 implies F(x1) > F(x2)

maximum

a function value f(a) is a relative -------- of f if there exists an interval (x1,x2,) containing a such that x1<x<x2 implies f(a)>=f(x)

average rate of change; secant

between any to points (x, f(x)) and (x2, f(x2)) is the slope of the line through the two points, this line is then called the -------- line

odd

a function f is ------- if for the each x in the domain of f, f(-x)=-f(x)

even

a function f is ------- if its graph is symmetric with respect to the y-axis.

f(x)=[IxI]

greatest integer function

f(x)=x

identity function

f(x)=1/x

reciprocal function

f(x)=x2

quadratic function

f(x)= square root of x

square root function

f(x)=c

constant function

f(x)= IxI

absolute value function

f(x)=x3

cubic function

f(x)=ax+b

linear function

rigid transformations

horizontal shifts, vertical shifts, and reflections

nonrigid

transformations that cause a distortion in the shape of the graph of y=f(x)

horizontal shrink; horizontal stretch

a nonrigid transformation of y=f(x) represented by h(x)=f(cx) is a ------------------ if c>1, and a -------------- if 0<c<1

vertical stretch; vertical shrink

a nonrigid transformation of y=f(x) represented by h(x)=cf(x) is a -------------------- if c>1, and a ----------------- if 0<c<1.

Vertical shift C units ^

h(x)=f(x)+c

vertical shift C units down

h(x)=f(x)-c

horizontal shift c units <---

h(x)=f(x+c)

horizontal shift c units --->

h(x)=f(x-c)

addition; subtraction; multiplication; division

two functions f & g can be combined by the arithmetic operations of ---, -------, -----, & ------ to create new functions.

product

the ----- of the function f with g is (f o g)(x)=f(g(x))

inner; outer

to decompose a function look for an --- and an ------- function

inverse

if the composite functions f(g(x))=x and g(f(x))=x then the function f is the ------ function of g

range; domain

the domain of f is the ------ of f-1, and the --------- of f-1 is the range of f

y=x

the graphs of f and f-1 are reflections of each other in the line -----.

one-to-one

a function is ------ if each value of the dependent variable corresponds to exactly one value of the independent variable

Horizontal Line Test

a graphical test for the existence of an inverse function of f

Ax+By+C=0

general form

x=a

vertical line

y=b

horizontal line

y=mx+b

slope-intercept form

y-y1=m(x-x1)

point-slope form

h(x)= -f(x); h(x)=f(-x)

a reflection in the x-axis of y=f(x) is represented by -------, while a reflection in the y-axis of y=f(x) is represented by -----------.

g(x)

the domain of (f o g) is all x in the domain of g such that ------ is in the domain of f.