Transformations in Functions
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Questions and Answers

What is the effect of adding a constant outside the function's expression?

  • It reflects the graph over the x-axis.
  • It moves the graph upwards. (correct)
  • It moves the graph left.
  • It moves the graph downwards.
  • What determines a horizontal shift in a function's graph?

  • Constants added or subtracted outside the function.
  • The degree of the polynomial function.
  • The coefficients of the function.
  • Constants added or subtracted inside the function. (correct)
  • In the function |x + 2|, how is the graph altered?

  • It is stretched vertically.
  • It is shifted left by two units. (correct)
  • It is shifted right by two units.
  • It is reflected over the y-axis.
  • Why are transformations useful in graphing functions?

    <p>They help avoid extensive point plotting.</p> Signup and view all the answers

    What happens when a constant is subtracted inside a function's expression?

    <p>The graph shifts right.</p> Signup and view all the answers

    Study Notes

    Transformations

    • Transformations are operations performed on functions to alter their graphs.
    • Transformations include shifts (up/down, left/right), stretches/compressions, and reflections.
    • Transformations streamline graphing, avoiding the need for extensive point plotting, especially for functions with infinite domains.

    Vertical Shift

    • Vertical shifts are determined by constants outside the function.
    • Adding a constant moves the graph up.
    • Subtracting a constant moves the graph down.
    • Changes affect the output values.

    Example of a Vertical Shift

    • x^2 + 3 shifts the parabola x^2 up 3 units.
    • The shift affects output values, moving the graph vertically.
    • Visualize by shifting the x-axis up while keeping the y-axis in place.

    Horizontal Shift

    • Horizontal shifts are determined by constants inside the function, typically within parentheses.
    • Adding inside the function shifts the graph to the left.
    • Subtracting inside the function shifts the graph to the right.
    • Changes affect the input values.

    Example of a Horizontal Shift

    • |x + 2 | shifts the absolute value function |x| left by 2 units.
    • The shift alters the input values, affecting the graph's horizontal position.
    • Visualize by shifting the y-axis to the left while keeping the x-axis in place.

    Key Takeaways

    • Transformations streamline graphing, preventing extensive point plotting.
    • Shifts: constants outside (vertical) or inside (horizontal) the function.
    • Visualization: imagine shifting the axes for easier plotting.
    • Memorize key points of basic functions for efficient transformations.

    Graph Transformations

    • Shifting Down: Negative constant after the function (e.g., f(x) - 1) shifts the graph down.
    • Shifting Left: Positive constant inside the function (e.g., f(x + 1)) shifts the graph left.
    • Shifting Right: Negative constant inside the function (e.g., f(x - 1)) shifts the graph right.
    • Vertical Stretch: Multiplying the output by a constant > 1 stretches vertically, making the graph narrower.
    • Vertical Compression: Multiplying the output by a constant between 0 and 1 compresses vertically, making the graph wider.
    • Horizontal Stretch: Multiplying the input by a constant between 0 and 1 stretches horizontally, making the graph wider.
    • Horizontal Compression: Multiplying the input by a constant > 1 compresses horizontally, making the graph narrower.
    • Reflection Across the X-axis: Multiplying the output by -1 reflects across the x-axis.
    • Reflection Across the Y-axis: Multiplying the input by -1 reflects across the y-axis.
    • Key Points: Understanding basic functions' key points allows for easier transformation application.
    • Symmetry: Even functions are symmetrical about the y-axis.
    • Square Root Graphs: The input of a square root function cannot be negative.

    Cube Root Function

    • The basic cube root function resembles an S-curve, flipped on its side.
    • Key points: (1, 1), (0, 0), and (-1, -1).
    • Reflecting across the x-axis (-∛x) changes key points to (1, -1), (0, 0), and (-1, 1).
    • Reflecting across the y-axis (∛(-x)) changes key points to (-1, 1), (0, 0), and (1, -1).

    Transformations Summary

    • Vertical Stretch/Compression: Impacts output, altering graph height.
    • Horizontal Stretch/Compression: Impacts input, altering graph width.
    • Reflection about x-axis: Changes sign of outputs, flipping over x-axis.
    • Reflection about y-axis: Changes sign of inputs, flipping over y-axis.
    • Vertical Shift: Adds a constant to output, shifting up/down.
    • Horizontal Shift: Adds a constant to input, shifting left/right.

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    Description

    This quiz covers the concept of function transformations, focusing on vertical shifts and how they affect the graph of a function. You'll learn how adding or subtracting constants outside a function alters its output values, using examples like the parabola. Understanding these transformations is crucial for efficient graphing of functions.

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