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Graphs and Functions (Review) MATH 207
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Distance Formula Example: Find distance between (-1,4) and (-4,-2). Answer: 6.71
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Midpoint Formula Example: Find the midpoint from P1(-5,5) to P2(-3,1). Answer: (-4,3)
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The standard form of an equation of a circle with radius r and center (h, k) is: The Unit Circle equation is: x y (h, k) r (x, y) Equations in two variables – Example: Circle Equations
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Definition of a Function
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Theorem: Vertical Line Test A set of points in the xy - plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.
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x y Not a function.
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x y Function.
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(a) For each x in the domain of f, there is exactly one image f(x) in the range; however, an element in the range can result from more than one x in the domain. (b) f is the symbol that we use to denote the function. It is symbolic of the equation that we use to get from an x in the domain to f(x) in the range. (c) If y = f(x), then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x. Summary Important Facts About Functions
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Properties of Functions: Even and Odd Functions A function f is even if for every number x in its domain the number -x is also in its domain and f(-x) = f(x) A function f is odd if for every number x in its domain the number -x is also in its domain and f(-x) = - f(x)
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Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.
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Where is the function increasing?
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Where is the function decreasing?
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Where is the function constant?
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Local Maxima and Minima Local Max
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Local Min
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Average rate of change of a Function
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From 0 to 1
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Library of Functions (Famous Functions)
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Piecewise-defined Functions: Example:
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Application problem:
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Graphing Functions:
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. The inverse of f, denoted by f -1, is a function such that f -1 (f( x )) = x for every x in the domain of f and f(f -1 (x))=x for every x in the domain of f -1 : Inverse Functions
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Theorem The graph of a function f and the graph of its inverse are symmetric with respect to the line y = x.
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y = x (2, 0) (0, 2)
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