Math 102
2.4 "More on Graphing"
Objectives:
* Graph nonlinear equations.
* Determine if the graph of an equation is symmetric with respect to the x axis, y axis, or the origin.
Preliminaries:
In Section 2.1, we learned to graph linear equations (straight lines) by constructing a table and by …nding the x intercept
and y intercept. In this section we will learn how to graph nonlinear equations.
It is very helpful to recognize that a certain type of equation produces a particular kind of graph. We need to develop
some general graphing techniques to use with equations when we do not recognize the graph.
Techniques to Graph Equations:
i: Find the intercepts.
ii: Solve the equation for y in terms of x or for x in terms of y.
iii: Set up a table of ordered pairs that satisfy the equation.
iv: Plot the points associated with the ordered pairs and connect them with a smooth curve.
y Axis Symmetry:
The graph of an equation is symmetric with respect to the y
if replacing x with
Example 1: (y axis symmetry)
a) Graph y = x2 + 2.
i. Intercepts
ii. Solve.
iii. Table of values
x
y
axis
x results in an equivalent equation.
iv. Plotting the Graph
(x; y)
y 12
10
8
6
4
2
-6
2
b) Show that y = x + 2 is symmetric about the y
-4
-2
-2
2
4
6
x
axis.
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Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.4
x Axis Symmetry:
kThe graph of an equation is symmetric with respect to the x
Example 2: (x
ii. Solve
iii. Table of values
y
y results in an equivalent equation.k
axis symmetry)
a) Graph y 2 x = 4.
i. Intercepts
x
axis if replacing y with
iv. Plotting the Graph
(x; y)
y
6
4
2
-5
5
10
-2
x
-4
-6
b) Show that y 2
x = 4 is symmetric about the x
axis.
Origin Symmetry:
The graph of an equation is symmetric with respect to the origin if replacing
x with
x and y with
y results in an equivalent equation.
Example 3: (Origin symmetry)
a) Graph y = x3 .
i.
Intercepts
iii. Table of values
x
y
ii. Solve.
iv. Plotting the Graph
y
(x; y)
10
5
-4
-2
2
4
x
-5
-10
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Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.4
b) Show that y = x3 is symmetric about the origin.
From the symmetry tests, we observe that if a curve has both x axis and y axis symmetry, then it must have
Note:
origin symmetry. However, it is possible for a curve to have origin symmetry and not be symmetric to
either axis. (Example 3)
Another graphing consideration is that of restricting a variable to ensure real number solutions. The following
example illustrates this point.
Example 4: (Sketching a graph by restricting a variable)
p
Graph y = x + 1
y
6
4
x
y
(x; y)
2
-2
2
4
6
8
10
x
-2
Now that we have the concept of symmetry we can add it to the list of graphing techniques.
Techniques to Graph Equations:
i:
Determine what type of symmetry the equation exhibits.
ii: Find the intercepts.
iii: Solve the equation for y in terms of x or for x in terms of y.
iv: Determine the restrictions necessary to ensure real number solutions.
[The type of symmetry and the restriction will a¤ ect our choice of values in the table.]
v: Set up a table of ordered pairs that satisfy the equation.
vi: Plot the points associated with the ordered pairs and connect them with a smooth curve.
[If appropriate, re‡ect this curve according to the symmetry possessed by the graph.]
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Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
2.4
Example 5: (Using the techniques )
Graph
i.
x2
y2 = 9
Symmetry
v. Table of values
x
ii. Intercepts
y
(x; y)
vi. Plotting the Graph
y
6
4
2
-6
-4
-2
2
-2
4
6
x
-4
-6
iii & iv. Solving / Restriction
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Notes by Bibiana Lopez