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Transformations of Graphs in College Algebra

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  • What is a vertical shift in graph transformations?
    A vertical shift moves the graph up or down by adding or subtracting a constant outside the function, as in \(f(x)+k\).
  • How does a horizontal shift affect a graph?
    A horizontal shift moves the graph left or right by adding or subtracting a constant inside the function's input, as in \(f(x-h)\).
  • What does vertical stretching or compressing mean?
    Multiplying the function by a constant >1 stretches it vertically; multiplying by a constant between 0 and 1 compresses it vertically.
  • How does horizontal stretching or compressing work?
    Replacing \(x\) with \(bx\) compresses the graph horizontally if \(b>1\), and stretches it if \(0<b<1\).
  • What effect does multiplying a function by -1 have?
    Multiplying by -1 reflects the graph across the x-axis, flipping it upside down.
  • What is the result of replacing \(x\) with \(-x\) in a function?
    It reflects the graph across the y-axis, creating a mirror image horizontally.
  • How do you write the transformation for shifting a graph 3 units up?
    Add 3 outside the function: \(f(x)+3\).
  • How do you represent shifting a graph 2 units to the right?
    Replace \(x\) with \(x-2\): \(f(x-2)\).
  • What transformation does \(y=2f(x)\) represent?
    A vertical stretch by a factor of 2, making the graph twice as tall.
  • What does \(y=f(\frac{x}{3})\) do to the graph?
    It horizontally stretches the graph by a factor of 3.
  • How does \(y=-f(x)\) transform the graph?
    It reflects the graph across the x-axis.
  • What is the effect of \(y=f(-x)\) on a graph?
    It reflects the graph across the y-axis.
  • How do you combine multiple transformations?
    Apply transformations in order: horizontal shifts and stretches inside the function first, then vertical stretches/compressions and shifts outside.
  • What is the general form for a transformed function combining shifts and stretches?
    \(y=a f(b(x-h))+k\), where a is vertical stretch/compression, b horizontal stretch/compression, h horizontal shift, and k vertical shift.
  • How does changing the sign of a in \(y=a f(x)\) affect the graph?
    If a is negative, the graph reflects across the x-axis.
  • What happens to the graph of \(f(x)\) when \(b>1\) in \(f(bx)\)?
    The graph compresses horizontally by a factor of \(\frac{1}{b}\).
  • What is the effect of \(0<b<1\) in \(f(bx)\)?
    The graph stretches horizontally by a factor of \(\frac{1}{b}\).
  • How do you identify a vertical compression from a function transformation?
    When the multiplier outside the function is between 0 and 1, the graph compresses vertically.
  • What is the effect of adding a constant inside the function argument, like \(f(x+4)\)?
    It shifts the graph horizontally 4 units to the left.