Transcript 3.2&3.3

3.2 Graphing Functions and
Relations
• Another way that we can determine if a
relation is a function is by looking at a graph of
the relation.
• A relation is a function if and only if no
vertical
line intersects the
graph more than once.
Graph the relation and then state whether or not
the relation is a function.
1.
0, 4 , 3,0, 1,2, 3,1, 4,0
Graph the relation and then state whether or not
the relation is a function.
2.
1,1, 1,2, 4,2, 5, 3, 2,3
Graph the relation defined by each of the following
rules over the domain 2,1,0,1,2, 3

3.
y5 x

Graph the relation defined by each of the following
rules over the domain 2,1,0,1,2, 3
4.
xy3


3.3 Graphing Linear Equations
Linear Equations
• Linear equations contain both an
• The graph of a linear equation is a
x
and a
line
y
.
• There are two forms of linear equations that we will
look at:
Slope-intercept
1.
, which is in the form
y  mx  b
2.
Standard
, which is in the form
Ax  By  C
Slope-Intercept Form
y  mx  b
• The m represents the slope of the line. The slope is
steepness
the
of the line.
– When the slope is positive, the graph goes up from
left to right.
– When the slope is negative, the graph goes down
from left to right.
rise
– When graphing, the slope can be thought of as
run
over
, where the numerator determines the
displacement and the denominator
vertical
horizontal
determines the
displacement
between points on the line.
Slope-Intercept Form
y  mx  b
• The b represents the
of the line.
y-intercept
In other words, it’s where the line crosses the
y-axis .
Determine the slope and y-intercept of
the lines below.
1.
1
y x4
2
2.
3y  x  12
Graphing in Slope-Intercept Form
• First, make sure that the equation is in the
form y=mx+b .
• Plot the y-intercept
first. Use this as your
starting point.
• Use the slope to plot another point, using the
idea of rise over run.
• Draw a line through your points with arrows
on each end.
Graph the equation.
3.
2
y x4
3
Graph the equation.
4.
y  4x  7
Graphing in Standard Form
• One way to graph from standard form is to
transform the equation into slope-intercept
form.
• A second way to graph in standard form is to
find the x-intercept and y-intercept. Plot the
two points and draw a line through them.
x-intercept
• The x-intercept is in the form  x , 0  . So to
find the x-intercept, you will need to
substitute 0 into the equation for y and
solve for x .
y-intercept
• The y-intercept is in the form  0 , y  . So to
find the x-intercept, you will need to
substitute 0 into the equation for x and
solve for y .
• You can use this method to find the x and yintercepts in slope-intercept form as well!
Find the x-intercept and y-intercept for
the following equations.
5.
3x  2y  12
6.
x y
 1
5 6
Graph using intercepts.
7.
5x  2y  10
Graph using intercepts.
8.
6x  3y  12
9.Determine the value of k so that the given
point will be on the graph of the given
equation.
4x  6k  3y;
3,10 
Vertical and Horizontal Lines
• The equation for a vertical line is in the form:
x = # . The slope of a vertical line is
undefined
.
• The equation for a horizontal line is in the
form: y = # . The slope of a horizontal line is
zero .
Graph the Equation
10.
y5
Graph the Equation
11.
x  3
Absolute Value Graphs
• Absolute value graphs are in the shape of a
V .
Use a table of values to graph the
function.
12.
y x
• On your graphing calculator graph the
following functions:
• Y1:
yx
• Y2:
y x3
• Y3:
y x4
• What happened to the graph?
• On your graphing calculator graph the
following functions:
• Y1:
yx
• Y2:
y  x  3
• Y3:
y  x  2 
2
2
2
• What happened to the graph?
• On your graphing calculator graph the
following functions:
• Y1:
y x
• Y2:
y x5 3
• Y3:
y  x 1  4
• What happened to the graph?
13. Without your calculator, graph the following
functions.
a.
y  x 1  2
13. Without your calculator, graph the following
functions.
y x2 3
b.