intercepts & symmetry
DESCRIPTION
By: Spencer Weinstein, Mary Yen, Christine Ziegler. Intercepts & Symmetry. 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 Fun ctions. Even Functions - PowerPoint PPT PresentationTRANSCRIPT
Intercepts & SymmetryBy: 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 FunctionsEven Functions
• Even functions are symmetric with respect to the y-axis. Essentially it’s y-axis symmetry.
Odd Functions• Odd Functions are symmetric with respect to
the origin. Essentially, it’s origin symmetry.
)()( xfxf
)()( xfxf
SymmetryX-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 symmetryAn 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
6)(2)(4 26 xxy
624 26 xxy
624 26 xxySubstitute “–x” for “x”
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
Substitute “–x” for “x” and “–y” for “y”
Simplify, Simplify, Simplify!
)()(2 3 xxy
xxy 32
xxy 32
xxy 32
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
24 )(7)(3 yyx
24 73 yyx
Substitute (-y) for y
Simplify, simplify, simplify!
24 73 yyx
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.
PracticeDoes this equation have y-axis symmetry?
235 xxy
2)()( 35 xxy
235 xxy
Substitute “–x” for “x”
Simplify, simplify, simplify!
No, because f(x) does not equal f(-x)
SymmetryThe 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?
1259
22
yx
Origin Symmetry
1259
22
yx
125)(
9)( 22
yx
1259
22
yx
Substitute “–x” for “x” and “–y” for “y”
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
1259
1259
)(
1259
22
22
22
yx
yx
yx
1259
125)(
9
1259
22
22
22
yx
yx
yxAs seen here, replacing “x” with“–x” will still yield the same equation. Therefore, his face has y-axis symmetry.
Replacing “y” with“–y” will still yield the same equation. Therefore, his face also has x-axis symmetry.
Even Mr. Spitz’s face is symmetrical!
(0, 5)
(0, -5)
(0, 3)(0, -3)
InterceptsY-intercept
The point(s) at which the graph intersects the y-axis
To find, let x = 0 and solve for yX-intercept
The point(s) at which the graph intersects the x-axis
To find, let y = 0 and solve for x
Finding x-interceptsxxy 43
xx 40 3
)2)(2(0 xxx
2,2,0 x
Let y = 0
Factor out an x
Solve equation for x
The x-intercepts are (-2,0), (0,0), and (2,0)
Finding y-intercepts
xxy 43 )0(4)0( 3 y
0y
The y-intercept is(0,0)
Let x = 0
Solve equation for y
Graph of
xxy 43
Y-axis and X-axis intercept
X-axis intercept
Mr. Spitz’s Snow ShopMr. Spitz sells snow for a living, and the sale
of his snow is modeled by the function where gives the
amount of snow in pounds at time x. Find the time at which Mr. Spitz needs to restock his snow.
23000722)( 2 xxxf)(xf
I’m an expert at it, too!
23000722)( 2 xxxf230007220 2 xx
)92)(2502(0 xx)2502(0 x
2502 x125x
)92(0 x
92x
Time is ALWAYS positive!
Mr. Spitz will need to restock his snow after 125 minutes.
Simplify, simplify, simplify!
X-axis intercept which x CANNOT equalX-axis intercept which x CAN equal
Yo yo
!
Come b
uy
some
snow
!
What a wonderful introductio
n to The Calculus!
We The Calculus!