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A function is Even if and only if its graph is symmetric with respect to the y - axis .

For every number x in its domain, the number – x is also in the domain and f ( x ) = f (– x ). For every number x in its domain, the number – x is also in the domain and f (– x ) . = – f ( x ).

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A function is Even if and only if its graph is symmetric with respect to the y - axis .

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  1. For every number x in its domain, the number –x is also in the domain and f(x) = f(– x). For every number x in its domain, the number –x is also in the domain and f(– x). = – f(x). A function is Even if and only if its graph is symmetric with respect to the y-axis. A function is Odd if and only if its graph is symmetric with respect to the origin.

  2. f(– x) = –(– x)4 + 3(– x )2 – 5 g(– x) = 5(– x)3 – (– x ) h(– x) = (– x)3 – 1 f(– x) = –x4 + 3 x2 – 5 g(– x) = – 5x3 +x h(– x) = – x3 – 1 f(– x) = f(x) g(– x) = – [5x3 –x] g(– x) = – g(x) f(x) is an Even function g(x) is an Odd function h(x) is neither function Even functions will always have even powers. Constants are with the even powers! Odd functions will always have odd powers.

  3. A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). Working defn. Reading a graph from left to right, as x-coord. are increasing the y- coord. are increasing. A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2). Working defn. Reading a graph from left to right, as x-coord. are increasing the y- coord. are decreasing. A function f is constant on an open interval I if, for any choice of x in I, the values of f(x) are equal. Working defn. Reading a graph from left to right, as x-coord. are increasing the y- coord. are constant. Forms a horizontal segment.

  4. MUST BE OPEN INTERVALS BY DEFINITION! X’s ONLY! -4 < x < -1 or (-4, -1) -6 < x < -4 or (-6, -4) -1 < x < 0 or (-1, 0) 3 < x < 6 or (3, 6) 0 < x < 3 or (0, 3)

  5. I I Y’s ONLY! (-1, 6 ) ( 0, 4 ) ( 3, 4 ) A function f has a local minimum at c if there is an open interval I containing c so that, for all xin I, f(x) >f(c). We call f(c) a local min. I I A function f has a local maximum at c if there is an open interval I containing c so that, for all xin I, f(x) <f(c). We call f(c) a local max. (-4, -4 ) (6, -5 ) A function f is defined on some interval I. If the is a number u in I for which f(x) > f(u) for all x in I, then f(u) is the absolute min. I I -4 , 4 Relative Minimums 6 , 4 Relative Maximums A function f is defined on some interval I. If the is a number u in I for which f(x) < f(u) for all x in I, then f(u) is the absolute max. -5 Absolute Minimum 6 Absolute Maximum

  6. Relative Minimum -1.531973 Relative Maximum 11.531973 Find the INTERVALS for where the graph is increasing and decreasing. Rounding errors with the x-coordinates. We will always write four decimal places. (– 0.8165, 0.8165) Decreasing (– , – 0.8165) (0.8165, ) Increasing ALWAYS OPEN INTERVALS( __ , __ )

  7. SLOPE 1st pt. (-1, 11) 2nd pt. (1, -1) 1st pt. (1, 8) 2nd pt. (3, 32)

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