Intercepts & Symmetry

By: Spencer Weinstein, Mary Yen, Christine Ziegler Intercepts & Symmetry Respect 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 Functions • Even 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.

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 Substitute “–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!

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 y 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.

PracticeDoes this equation have y-axis symmetry? 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?

Origin Symmetry 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 As 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.

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

Graph of Y-axis and X-axis intercept X-axis intercept

Mr. Spitz’s Snow Shop I’m an expert at it, too! Mr. 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.

Simplify, simplify, simplify! Mr. Spitz will need to restock his snow after 125 minutes. Time is ALWAYS positive!

X-axis intercept which x CANNOT equal X-axis intercept which x CAN equal

Yo yo! Come buy some snow!

What a wonderful introduction to The Calculus! We The Calculus!

Intercepts & Symmetry