March 28, 2016
Math 95
DAY 1
Sections 3.1, 3.2
Section 3.1: Graphing Equations
You should be familiar with plotting points, ordered pairs, etc.
But let us review some plotting and our quadrants.
Practice:
Plot each ordered pair and
name the quadrant or axis in
which the point is located.
a. (3, -2)
b. (0, 3)
c. (-4, 1)
d. (-1, 0)
e. (-2.5, -3)
f. (3.5, 4.5)
(x, y) it is
alphabetical!
Determining Whether an Ordered Pair is a Solution
Solutions of equations are ordered pairs that form a true statement when
substituted into the equation.
(x, y) it is
alphabetical!
Practice: Determine whether (0, 6) and (1, 4) are solutions of
the equation 2x + y = -6
x
y
Can you find another solution to the
above equation?
The previous equation 2x + y = -6 is an example of a linear equation.
To graph a linear equation we only need two solutions/ordered pairs
(two points make a line).
There are infinitely many solutions.
Every ordered pair solution corresponds to a point on the line.
Every point on the line corresponds to an ordered pair solution.
Practice: Graph 2x + y = -6
"What were two of our solutions
from the previous slide?"
March 28, 2016
Linear Equation in Two Variables
is any equation that can be written in the form Ax + By = C
Standard Form
the graph of a linear equation is a line.
Graph: 2x + 3y = 6
To graph this equation we create a
table of values and identify ordered
pairs (find two or more solutions).
An easy way to find two points to plot
are to find the x- and y- intercepts.
"Plugging in zero is easy!"
Finding x- and y- intercepts
To find the x- intercept, let Y = 0 and solve for x.
To find the y- intercept, let x = 0 and solve for y.
Non-linear Equations
Not all equations are linear.
Not all graphs of two variables are lines.
We can still graph these equations we just need more points to sketch.
There are many types of non-linear equations:
quadratic, exponential, square root, absolute value, etc.
It will really help you if you have some idea what each kind of
equation looks like (ex. quadratic equations are parabolas).
If it cannot be written in the form y = ax + b or AX + By = C,
then it is non-linear!
Quadratic
y = x2 + 3
Parabola
x
-3
-2
-1
0
1
2
3
y
March 28, 2016
Absolute Value
x
y
-10
-5
-2
0
2
5
10
How to sketch the graph of a non-linear equation:
*Make a table of values.
*Look for a pattern.
Examine this earlier example.
While making a table of values you might have noticed that the
smallest value this equation could possibly output, occurs with an
input of 0. Why?!
If you are unsure about the graph of a nonlinear equation, find
more ordered pair solutions!
Section 3.2 Introduction to Functions
Vocabulary:
Ordered Pairs: (x, y) represents a point in the plane in
terms of distance along each axis from the origin.
Relation: is a set of ordered pairs.
Domain: set of all first components of the ordered pairs (x).
Range: set of all second components of the ordered pairs (y).
Practice: Determine the domain and range of the relation.
{(1,6), (2,8), (0,3), (0,-2)}
*Domain or range elements that occur more than once need to be listed only once.
March 28, 2016
A function is a relation in which each first component (input) in the
ordered pairs corresponds to exactly one second component (output).
Vertical Line Test
Graphs can be used to help determine
whether a relation is also a function.
If no vertical line can be drawn so that
it intersects a graph more than once,
the graph is the graph of a function.
One Red Input
Three Green Outputs
Determine whether the following relations are functions
Function notation, f(x) is just a fancy name for y.
We only use it if we know that we have a function rather than just a
relation.
It means f is a function of x, x is the independent variable and y, or
f(x), is the dependent variable.
f(x) looks like
multiplication of f and x,
but it is NOT!
Instead it immediately
gives important
information about the
relation.
March 28, 2016
Find the domain and range of these relations. Are they functions?
See how much quicker it is to say,
"find f(2)"
rather than saying,
"find y when x is equal to 2".
We can also use a graph to find a function value.
Find:
f(2) =
f(-2) =
f(0) =
Is it a problem that f(2) = f(-2) = 3?