Skip to main content
Back

Precalculus: Function Behavior and Transformations

Control buttons has been changed to "navigation" mode.
1/23

Terms in this set (23)

  • What is a function?
    A function is a mathematical relation that maps each input value to exactly one output value.
  • What is the domain of a function?
    The domain is the set of all input values of a function, represented by the x variable.
  • What is the range of a function?
    The range is the set of all output values of a function, represented by the y variable.
  • When is a function positive or negative based on its graph?
    A function is positive when its graph lies above the x-axis (outputs > 0), and negative when it lies below the x-axis (outputs < 0).
  • What are the four mathematical representations of functions?
    Functions can be represented graphically, analytically (equations), numerically (tables), and verbally.
  • What does it mean for a function to be increasing on an interval?
    A function is increasing if as input values increase, output values also increase: if \(a
  • What does it mean for a function to be decreasing on an interval?
    A function is decreasing if as input values increase, output values decrease: if \(af(b)\).
  • What is the difference between a function increasing and its rate of change increasing?
    A function increasing means outputs rise as inputs rise; the rate of change increasing means the slope itself is increasing (function is concave up).
  • What is the meaning of 'rate of change' in this course?
    'Rate of change' means the slope of the function. It describes how output values change relative to input values.
  • What is a point of inflection on a graph?
    A point of inflection is where the graph changes concavity, from concave up to concave down or vice versa.
  • What are vertical and horizontal asymptotes?
    Vertical asymptotes are lines where the function approaches infinity; horizontal asymptotes describe the end behavior of the function as input approaches infinity or negative infinity.
  • What does it mean for a function to be even or odd?
    An even function satisfies \(f(-x)=f(x)\); an odd function satisfies \(f(-x)=-f(x)\).
  • What is a one-to-one function?
    A function is one-to-one if each output corresponds to exactly one input; its inverse is also a function.
  • What is the vertical line test used for?
    The vertical line test determines if a graph represents a function by checking if any vertical line intersects the graph more than once.
  • How do vertical translations affect a function graph?
    A vertical translation shifts the graph up or down by \(k\) units: \(g(x)=f(x)+k\).
  • How do horizontal translations affect a function graph?
    A horizontal translation shifts the graph left or right by \(h\) units: \(g(x)=f(x+h)\) shifts left by h.
  • What is vertical dilation of a function?
    Vertical dilation stretches or compresses the graph vertically by a factor \(a\): \(g(x)=af(x)\).
  • What is horizontal dilation of a function?
    Horizontal dilation stretches or compresses the graph horizontally by a factor \(1/b\): \(g(x)=f(bx)\).
  • What happens when \(a<0\) in vertical dilation?
    If \(a<0\), the graph is reflected across the x-axis.
  • What happens when \(b<0\) in horizontal dilation?
    If \(b<0\), the graph is reflected across the y-axis.
  • How do transformations inside the function argument affect the graph?
    Transformations inside the parentheses affect the x-values and do the opposite of what they look like (e.g., \(x+h\) shifts left).
  • How do transformations outside the function argument affect the graph?
    Transformations outside the parentheses affect the y-values and do exactly what they look like (e.g., \(f(x)+k\) shifts up by k).
  • What is the relationship between a function and its inverse graphically?
    The graph of a function and its inverse are reflections of each other over the line \(y=x\).