Transformations in Functions

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.