Math 140
1.2: Intercepts; Symmetry; Graphing Key Equations; Circles
Trigonometry
In this section we need to be able to:
1. Compute x and y-intercepts, p.17
2. Work with symmetry
o Determine symmetry visually
o Complete a graph using symmetry
o Test for symmetry algebraically, p. 19
3. Use a graphing calculator to compute the x and y-intercepts (this was in the 4th edition)
To Find the x–intercept:
Computing the x and y–intercepts
To Find the y–intercept:
Let y = 0, and solve for x.
Note: x–intercepts will always look like  ( , 0)
Symmetry with
respect to
the x–axis
If (x, y) is on the graph, then
(x, –y) is on the graph.
Let x = 0, and solve for y.
Note: y–intercepts will always look like  (0, )
Symmetry with
respect to
the y–axis
If (x, y) is on the graph, then
(–x, y) is on the graph.
Symmetry with
respect to
the Origin
If (x, y) is on the graph, then
Graphical
(–x, –y) is on the graph.
Condition
1.) reflection in the y-axis,
x–axis acts a as mirror
y–axis acts as a mirror
2.) followed by reflection in
Graphical
(reflection in the x–axis)
(reflection in the y–axis)
the x–axis
Interpretation
or visa versa …
Replace x with –x , AND
Replace y with –y
Replace x with –x
y with –y
Test
Condition
If an equivalent equation results, the graph has the desired symmetry.
Hint  Think OPPOSITE
Symmetry with respect
to the x–axis
Symmetry with respect
to the y–axis
y x40
y  x 1
2
4
Symmetry with respect
to the Origin
y
x4 1
2x5
Example 1: Complete the graph so that it has the type of symmetry indicated.
x–axis
y–axis
Origin
Example 2: (62) List the intercepts and test for symmetry.
 You should be able to verify the symmetry visually on your graphing calculator; then
 compute the intercepts with the correct key punch sequence
y
x2  4
2x
Directions for computing the intercepts on the TI Graphing Calculator are on your graphing calculator handouts.
Why do we need to know about symmetry?
Knowledge about symmetry makes graphing easier since we only need to do half of the work. For example,
suppose that we know a graph is symmetric with respect to the y–axis. Then the y–axis acts as a mirror
reflecting quadrants I and IV (the right plane) onto quadrants II and III (the left plane), or visa versa. So when
we set up our tables of values, we only need to consider x > 0, because when x < 0 the y–values produced are
the same.
Why do we need to know how to calculate the x and y–intercepts by hand?
Often times we are given an equation (and no other information) and asked to draw a graph of the equation.
Even if we are using a graphing calculator this can still be a time consuming task because we need to get the full
graph (all the important details) in the viewing rectangle of the graphing calculator. The standard viewing
rectangle that we use is –10 to +10 along the x and y–axes. Often times this “WINDOW” will show the full
graph, but sometimes it will not. When this happens knowledge about the x and y–intercepts will allow us to
adjust the WINDOW the correctly position all the important details of the graph in the viewing rectangle.