By: Spencer Weinstein, Mary Yen, Christine Ziegler
Respect The
Calculus!
Students Will Be Able To
identify different types of symmetry and
review how to find the x- and y- intercepts of an equation.
Even/Odd Functions
 Even Functions
• Even functions are symmetric with respect to the y-axis.
Essentially it’s y-axis symmetry.
f (  x)  f ( x)
 Odd Functions
• Odd Functions are symmetric with respect to the
origin. Essentially, it’s origin symmetry.
f (  x)   f ( x)
Symmetry
 X-axis symmetry
 An equation has x-axis symmetry if replacing the “y” with a
“-y” yields an equivalent equation.
 The graph should look the same above and below the x-axis.
 Y-axis symmetry
 An equation has y-axis symmetry if replacing the “x” with a
“-x” yields an equivalent equation.
 The graph should look the same to the left and right of the
y-axis.
 Origin symmetry
 An equation has origin symmetry if replacing the “x” with a
“-x” and “y” with a “-y” yields an equivalent equation.
 The graph should look the same after a 180° turn.
Y-axis Symmetry Practice
y  4x  2x  6
6
2
Substitute “–x” for “x”
y  4( x)  2( x)  6
6
2
y  4x  2x  6
6
2
Simplify, simplify, simplify!
Since the equation is the same as the initial after “x” was
replaced with “-x,” the equation must have y-axis
symmetry. In addition, that would mean that it is an
even function.
Origin Symmetry Practice
y  2x  x
3
Substitute “–x” for “x” and “–y” for “y”
 y  2( x)  ( x)
3
Simplify, Simplify, Simplify!
 y  2 x  x
3
y  2x  x
3
Since the equation is the same as the initial after “x” was
replaced with “-x,” and “y” was replaced with “-y,” the equation
must have origin symmetry. In addition, that would mean
that it is an odd function.
X-axis Symmetry Practice
x  3y  7 y
4
2
Substitute (-y) for y
x  3( y)  7( y)
4
x  3y  7 y
4
2
2
Simplify, simplify, simplify!
Since the equation is the same as the initial after “y” was replaced with “-y,”
the equation must have x-axis symmetry.
Graph of x-axis symmetry
The graph to the left
exemplifies x-axis
symmetry. However, note
that it’s not the graph of the
equation listed above.
Practice
Does this equation have y-axis symmetry?
y  x  x 2
5
3
Substitute “–x” for “x”
y  ( x)  ( x)  2
5
3
y  x  x  2
5
Simplify, simplify, simplify!
3
No, because f(x) does not equal f(-x)
Symmetry
 The following equation gives the general shape of Mr.
Spitz’s face. Does Mr. Spitz have y- and/or x-axis
symmetry? How about origin symmetry?
2
2
x
y

1
9 25
Origin Symmetry
2
2
x
y

1
9 25
Substitute “–x” for “x” and “–y” for “y”
( x)
( y )

1
9
25
2
2
2
2
x
y

1
9 25
Simplify, Simplify, Simplify!
The result is identical to the initial
equation. Therefore, Mr. Spitz’s face has
origin symmetry.
Y-axis and X-axis Symmetry
2
2
x
y

 1 As seen here,
replacing “x”
9 25
with“–x” will still
2
2
yield the same
( x)
y

 1 equation.
9
25 Therefore, his
face has y-axis
2
2
symmetry.
x
y

1
9 25
Replacing “y”
with“–y” will still
yield the same
equation.
Therefore, his
face also has xaxis symmetry.
x2 y2

1
9 25
x 2 ( y ) 2

1
9
25
2
2
x
y

1
9 25
(0, 5)
Even Mr.
Spitz’s face is
symmetrical!
(0, -3)
(0, 3)
(0, -5)
Intercepts
 Y-intercept
 The point(s) at which the graph intersects the y-axis
 To find, let x = 0 and solve for y
 X-intercept
 The point(s) at which the graph intersects the x-axis
 To find, let y = 0 and solve for x
Finding x-intercepts
y  x  4x
3
0  x  4x
Let y = 0
3
Factor out an x
0  x( x  2)(x  2)
x  0,2,2
Solve equation for x
The x-intercepts are
(-2,0), (0,0), and (2,0)
Finding y-intercepts
y  x  4x
3
y  (0)  4(0)
3
y  0
Let x = 0
Solve equation for y
The y-intercept is
(0,0)
Graph of
y  x  4x
Y-axis and X-axis intercept
X-axis intercept
3
Mr. Spitz’s Snow Shop
 Mr. Spitz sells snow for a living, and the sale of his snow
f ( x)  2x 2  72x  23000
is modeled by the function
where f (x) gives the amount of snow in pounds at
time x. Find the time at which Mr. Spitz needs to
restock his snow.
Mr. Spitz will need to
restock his snow after 125
minutes.
f ( x)  2x  72x  23000
0  2 x 2  72x  23000
0  (2 x  250)(x  92)
0  ( x  92)
2
0  (2 x  250)
2 x  250
x  92
x  125
Time is ALWAYS positive!
X-axis intercept which x CANNOT equal
X-axis intercept which x CAN equal
What a
wonderful
introduction
to The
Calculus!
We
The Calculus!