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L4.3 Reflecting Graphs and Symmetry

L4.3 Reflecting Graphs and Symmetry. Reflections are something you do to a function You reflect a function over a line Symmetry is an attribute that a function may possess A graph may be symmetric wrt a line or a point. Warmup. Working in pairs, complete Activities #1-4 on p131 of textbook

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L4.3 Reflecting Graphs and Symmetry

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  1. L4.3 Reflecting Graphs and Symmetry • Reflections are something you do to a function • You reflect a function over a line • Symmetry is an attribute that a function may possess • A graph may be symmetric wrt a line or a point

  2. Warmup Working in pairs, complete Activities #1-4 on p131 of textbook You may use a graphing calculator For each question, you will generate an X/Y grid for each part, a and b. So 8 grids all together. Each grid will have 2 curves. Graph the one using a dashed line and the other using a solid line.

  3. y = –f(x) y = f(x) y = f(x) y = |f(x)| Reflections of Graphs • Graphs of equations can be reflected over a line. • If y = f(x) is reflected over the x-axis, then for every x, the opposite value for y is obtained, i.e., y = –f(x). • If the absolute value of f(x) is taken, i.e., y = |f(x)|, then the graph is reflected (i.e., y = –f(x)) whenever f(x) < 0. x-Axis Reflection This is a vertical reflection. Abs Value “Reflection”

  4. y = f(x) y = f(–x) y = f(x) x = f(y) Reflections • If y = f(x) is reflected over the y-axis, then the same value of y is obtained for –x, i.e., y = f(–x). • If y = f(x) is reflected over the line y = x, then all x’s and y’s are interchanged. y-Axis Reflection This is a horizontal reflection. Reflection over y = x line

  5. Symmetry • Reflection is something you do to an equation or function. • Symmetry is an attribute of an equation or function that can assist in graphing. • Graphs of equations can have symmetry with respect to a line (axis or symmetry) or a point (point of symmetry). • We will study symmetry wrt: • x-axis, • y-axis • the origin • the identity function, y = x These are the important onesin terms of functions….

  6. x-Axis Symmetry x-Axis Symmetry for every (x, y) on the graph, (x, –y) is alsoon the graph. To test: Replace all y’s with –y. Does an equivalent equation result? Test x + y2 = 7 for x-axis symmetry. These eqns are not functions since they fail the vertical line test.

  7. These eqns are even functions. y-Axis Symmetry y-Axis Symmetry for every (x, y) on the graph, (–x, y) is alsoon the graph. To test: Replace all x’s with –x. Does an equivalent equation result? Test y = x4 – x2 + 3 for y-axis symmetry.

  8. These eqns are odd functions. Origin Symmetry Origin Symmetry for every (x, y) on the graph, (– x, –y) is alsoon the graph. To test: Replace all x’s with –x and all y’s with –y. Does an equivalent equation result? Test y = x3 + x for origin symmetry.

  9. Line y = x Symmetry Line y = x Symmetry for every (x, y) on the graph, (y, x) is alsoon the graph. (x , y) (y , x) To test: Interchange x and y. Does an equivalent equation result? Test x + y = 5 for line y = x symmetry. Equations w/ y = x symmetry are not extensively studied. Instead, if the graphs of two functions are symmetric wrt the y = x line, then they are inverse functions (Lesson 4.5)

  10. Symmetry Examples Some equations can have more than one type of symmetry. What type of symmetry does x2 + y2 = 1 have?

  11. Even and Odd Functions • Even and Odd refer only to functions • Most functions are neither even nor odd. • y = f(x) is even if in the domain of f, (symmetric wrt y-axis) • y = f(x) is odd if in the domain of f, (symmetric wrt origin) • Otherwise, y = f(x) is neither even nor odd. • To test if a function is even or odd, replace –x for x and determine if the fcn is the same (even), or opposite (odd), or neither. • A function can not be both even and odd. Why?

  12. Symmetry of Functions • Most functions do not have any symmetry • When they do, we study • y-axis (even), or • origin (odd) • Not studied: • x-axis Does not pertain to functions • y = x Not studied as a function attribute; Instead, we study 2 functions that are symmetric wrt y = x (inverses)

  13. Shifted Quadratics & Cubics (neither even nor odd) • We already know how to find the “middle” (axis of symm) of a quadratic function: ax2 + bx + c • To find the “middle” (turning point or point of symmetry) of a cubic function: ax3 + bx2 + cx + d • Cubics are odd functions so they are symmetric over (h, k). • If there is a point (x1, y1) on the curve, there is a corresponding point (x2, y2) on the other side such that the point of symmetry is in the middle; i.e.,

  14. Example P137 #30 y = –x3 – 6x2 – 9x has a local minimum at (–3, 0). Find the point of symmetry and then deduce the coordinates of the local maximum. Point of symmetry for cubics: Cubics have point symmetry, so the corresponding point to the local max @ (–3, 0) is a local min @ (x2, y2) on the other side of (–2, 2). The point of symmetry, (–2, 2), is in the middle of the max and min points. So, the local min, (x2, y2) = (–1, 4)

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