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College Algebra: Functions and Graphs

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이 집합의 용어 (22)

  • What does it mean for a function to be increasing on an interval?
    A function is increasing on an interval if the function values go up from left to right over that interval.
  • What does it mean for a function to be decreasing on an interval?
    A function is decreasing on an interval if the function values go down from left to right over that interval.
  • What does it mean for a function to be constant on an interval?
    A function is constant on an interval if the function values do not change (neither increase nor decrease) from left to right over that interval.
  • Define a relative (local) maximum of a function.
    A function has a relative maximum at a point if the y-value there is larger than all other y-values near that point.
  • Define a relative (local) minimum of a function.
    A function has a relative minimum at a point if the y-value there is smaller than all other y-values near that point.
  • What is x-axis symmetry in a relation?
    A relation has x-axis symmetry if for every point (x,y), the point (x,-y) is also on the graph.
  • What is y-axis symmetry in a relation?
    A relation has y-axis symmetry if for every point (x,y), the point (-x,y) is also on the graph.
  • What is origin symmetry in a relation?
    A relation has origin symmetry if for every point (x,y), the point (-x,-y) is also on the graph.
  • Algebraic test for x-axis symmetry?
    Replace y with -y in the equation. If the equation remains unchanged, the relation has x-axis symmetry.
  • Algebraic test for y-axis symmetry?
    Replace x with -x in the equation. If the equation remains unchanged, the relation has y-axis symmetry.
  • Algebraic test for origin symmetry?
    Replace x with -x and y with -y in the equation. If the equation remains unchanged, the relation has origin symmetry.
  • Algebraic definition of an even function?
    A function f is even if \(f(-x) = f(x)\) for all x in its domain.
  • Algebraic definition of an odd function?
    A function f is odd if \(f(-x) = -f(x)\) for all x in its domain.
  • Can a function have x-axis symmetry?
    Virtually no functions have x-axis symmetry because that would fail the vertical line test.
  • What happens when you add or subtract even functions?
    The result is always an even function.
  • What happens when you add or subtract odd functions?
    The result is always an odd function.
  • What is a piecewise function?
    A function defined by two or more different expressions over different parts of its domain.
  • What is the difference quotient of a function f?
    The difference quotient is \(\frac{f(x+h) - f(x)}{h}\), where \(h \neq 0\).
  • Why is the difference quotient important?
    It measures the average rate of change of the function over the interval from x to x+h and is foundational for derivatives.
  • How to determine intervals where a function is increasing or decreasing from a graph?
    Look for intervals where the graph moves upward (increasing) or downward (decreasing) as you move left to right.
  • How to identify relative maxima and minima from a graph?
    Relative maxima are peaks where the graph changes from increasing to decreasing; minima are valleys where it changes from decreasing to increasing.
  • How to determine if a function is even, odd, or neither from its graph?
    Check for y-axis symmetry (even), origin symmetry (odd), or lack of both (neither).