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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!