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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 Presentation

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Intercepts & Symmetry

Intercepts & Symmetry

By: Spencer Weinstein, Mary Yen, Christine ZieglerRespect The Calculus!Students Will Be Able To identify different types of symmetry andreview how to find the x- and y- intercepts of an equation.

Even/Odd FunctionsEven FunctionsEven functions are symmetric with respect to the y-axis. Essentially its y-axis symmetry.

Odd Functions Odd Functions are symmetric with respect to the origin. Essentially, its origin symmetry.

SymmetryX-axis symmetryAn 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 symmetryAn 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

Substitute x for xSimplify, 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 ySimplify, Simplify, Simplify!

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

Substitute (-y) for ySimplify, 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 its not the graph of the equation listed above.

PracticeDoes this equation have y-axis symmetry?

Substitute x for xSimplify, simplify, simplify!

No, because f(x) does not equal f(-x)SymmetryThe following equation gives the general shape of Mr. Spitzs face. Does Mr. Spitz have y- and/or x-axis symmetry? How about origin symmetry?

Origin Symmetry

Substitute x for x and y for ySimplify, Simplify, Simplify!The result is identical to the initial equation. Therefore, Mr. Spitzs face has origin symmetry.

Y-axis and X-axis Symmetry

As seen here, replacing x withx will still yield the same equation. Therefore, his face has y-axis symmetry.Replacing y withy will still yield the same equation. Therefore, his face also has x-axis symmetry.

Even Mr. Spitzs face is symmetrical!(0, 5)(0, -5)(0, 3)(0, -3)InterceptsY-interceptThe point(s) at which the graph intersects the y-axisTo find, let x = 0 and solve for yX-interceptThe point(s) at which the graph intersects the x-axisTo find, let y = 0 and solve for x

Finding x-intercepts

Let y = 0Factor out an xSolve equation for x

The x-intercepts are (-2,0), (0,0), and (2,0)Finding y-intercepts

The y-intercept is(0,0)Let x = 0Solve equation for y

Graph of

Y-axis and X-axis intercept

X-axis intercept

Mr. Spitzs 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.

Im an expert at it, too!

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 buy some snow!

What a wonderful introduction to The Calculus!

We The Calculus!