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)

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?

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-intercepts

xxy 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 xxxf

230007220 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 equal

X-axis intercept which x CAN equal

Yo yo

!

Come

buy

som

e

snow

!

What a wonderful introductio

n to The Calculus!

We The Calculus!