Section 2.2
Graphing Equations: Point-Plotting,
Intercepts, and Symmetry
Graphing Equations by Plotting Points
The graph of an equation in two
variables, x and y, consists of all the
points in the xy plane whose
coordinates (x,y) satisfy the
equation.
Example
Does the point (-1,0) lie on the graph y = x3 – 1?
0   1  1
3
0  1  1
0  2
No
Graphing an Equation of a Line by
Plotting Points
Graph the equation: y = 2x-1
x
-2
-1
0
1
2
y=2x-1
(x,y)
Graphing a Quadratic Equation by
Plotting Points
Graph the equation: y=x²-5
x
-2
-1
0
1
2
y=x²-5
(x,y)
Graphing a Cubic Equation by Plotting
Points
Graph the equation:
y=x³
x
-2
-1
0
1
2
y=x³
(x,y)
X and Y Intercepts


An x–intercept of a graph is a point
where the graph intersects the xaxis.
A y-intercept of a graph is a point
where the graph intersects the yaxis.
Find the x and y intercepts.
x-intercepts:
(1,0) (5,0)
y-intercept:
(0,5)
What are the x and y intercepts of this
graph given by the equation:
y=x³-2x²-5x+6
x-intercepts:
(-2,0)(1,0)(3,0)
y-intercept:
(0,6)
How do we find the x and y intercepts algebraically?
First let’s examine the x-intercepts.




For example: The graph to
the right has the equation
y=x²-6x+5.
What is the y-coordinate
for both x-intercepts?
Zero.
So to find x intercepts we
can plug in zero for y and
solve for x:





0=x²-6x+5
0=(x-5)(x-1)
x-5=0 x-1=0
x=5,1
The x-intercepts are (1,0)
and (5,0)
Next, let’s find the y-intercept.




Equation: y=x²-6x+5.
What is the x-coordinate
for the y-intercept?
Zero.
So to find the y-intercept
we can plug in zero for x
and solve for y:



y=0²-6(0)+5
y=5
The y-intercept is (0,5)
Symmetry


The word symmetry conveys
balance.
Our graphs can be symmetric with
respect to the x-axis, y-axis and
origin.



This graph is
symmetric with
respect to the x-axis.
Notice the
coordinates: (2,1)
and (2,-1).
The y values are
opposite.



This graph is
symmetric with
respect to the y-axis.
What do you notice
about the
coordinates of this
graph?
The x values are
opposite.



This graph is
symmetric with
respect to the
origin.
What do you
notice about the
coordinates (2,3)
and (-2,-3)?
Both the x values
and y values are
opposite.
Summary

If a graph is symmetric about the…



X-axis, the y values are opposite
Y-axis, the x values are opposite
Origin, both the x and y values are
opposites
Testing for Symmetry with respect to
the x-axis
Test the equation y²=x³
Solution:
 Replace y with –y
 (-y)²=x³
 y²=x³

The equation is the same therefore it is
symmetric with respect to the x-axis.
Testing from symmetry with respect to
the y-axis
Test the equation y²=x³
Solution:
 Replace x with –x
 y²=(-x)³
 y²=-x³
 The equation is NOT the same therefore it
is NOT symmetric with respect to the yaxis.
Testing for Symmetry with respect to
the origin

Test the equation y²=x³
Solution:
 Replace x with –x and replace y with -y
 (-y)²=(-x)³
 y²=-x³
 The equation is NOT the same therefore it
is NOT symmetric with respect to the
origin.
Test for Symmetry: y = x5 + x

Y-axis: x changes to –x



Y = (-x)5 + -x
y = -(x5 + x)
No!

X-axis: y changes to –y



-y = x5 + x
y = -(x5 + x)
No!

Origin: y changes to –y and x
changes to –x




-y = (-x)5 + -x
-y = -(x5 + x)
y = x5 + x
Yes!