Download Graphs of functions, and algebra

Graphs of functions, and algebra
FUNCTIONS: As you remember, a function is
a rule that takes in one number and gives out
another.
It’s a special case of a relation: a relation
is a function if every element in the domain
relates to precisely one element of the
codomain, so it makes sense to talk about
“the element f(x) related to x”
A partial function has each element in the
domain related to at most one element in the
domain. We can talk about “the element, if
there is one , related to x”
NOTATION: a reminder
If we introduce a dummy variable in
referring to a function, it needs to do
something.
Rigorous: "the sine function"
"the function mapping x to sin(x)"
“the function mapping x to x3+3x-1”
Casual: "The function y = sin(x)" (depends
on a convention that "x" is usually the domain
and “y” the codomain; this can be tricky when
talking about inverses, composites, and
implicit functions (see below))
Sloppy: "the function sin(x)" (the role of
x is never made clear)
MORE NOTATION:
“The equation y = sin(x)” is fine. An
equation may involve any number of variables
and does not distinguish between “domain”
and “codomain”.
IF an equation implies a unique y value for
every x, or a unique x value for every y, these
are called “implicit functions”.
x3 = y5 generates an implicit function
x  x3/5 and an implicit function y y5/3.
x2 =y2 does not generate an implicit
function (unless we restrict the domain and
codomain).
GRAPHS
A relation from the real numbers to the
real numbers can (if it's simple enough) be
represented by a graph.
Perpendicular (horizontal and vertical)
axes are drawn.
Every point in the plane is given real
coordinates (x,y) representing horizontal and
vertical distance.
The plane is thus the product of two
copies of the real numbers.
Every subset of the plane represents a
relation R  R.
This is exactly like a relation table but for
infinitely many values.
Draw and label a pair of axes.
Plot the points (0,0), (0,1), (0,2) with a
solid dot.
Plot the points (1,0), (2,0), (-1,0) with a
small ring.
Plot the point (-1,1) with a star.
A square has two opposite corners at (1,2)
and (-1,-2). Draw the square and find the
coordinates of the other two vertices.
Draw and compare on (a) the reals and (b) the
set {0,1,2,3,4}:
(1) the relation x=y
(2) the relation x>y
(3) the function or partial function taking
x to x+1
(4) the function or partial function taking
x to x2
THE VERTICAL LINE TEST:
A set of points in the plane is the graph
of a function if every vertical line meets the
set in exactly one point.
A table represents a function if every
column contains exactly one entry.
A set of points in the plane is the graph
of a partial function if every vertical line
meets the set in at most one point.
A table represents a partial function if
Draw a set of axes and plot some points (x,y)
for which y=1. Guess what the set of all such
points is, and draw as much of it as you can.
Does the y value for one of these sets
depend on x?
Does this graph describe a relation ?
Does this graph describe a function ?
On the same axes, graph y=2 and y=-1.
Draw a set of axes and plot some points (x,y)
for which y = x+1. Then sketch in the set of
all such points.
Do the same thing for y = x and y=x-2.
What do they have in common?
Draw a set of axes and plot some points (x,y)
for which y = x+1. Then sketch in the set of
all such points.
Do the same thing for y = 2x+1 and
y= 1-x.
What do they have in common?
Is it true that every straight line is the
graph of a function?
Which straight lines are the graphs of
equations of the form
y = mx + b ?
What do these lines have in common?
The value m represents the slope. Positive
slopes rise from left to right, negative slopes
sink from left to right .
I
The value b represents the y-intercept – the
point at which x=0.
If the x-intercept is the point (if any) at
which y=0, what are its coordinates?
Which lines don’t have x-intercepts?
Plot the relations:
x+y=1
x + 2y = 1
x - 2y = 2
3x + 2y = 6
Are these functions?
Can you guess the slope and intercepts of
2x + 3y = 12 ;
x–y=2 ?
Check your answers by plotting.
The slope of ax + by = c is:
The x-intercept is:
The y-intercept is: