Podcast
Questions and Answers
What is the effect of adding a constant outside the function's expression?
What is the effect of adding a constant outside the function's expression?
What determines a horizontal shift in a function's graph?
What determines a horizontal shift in a function's graph?
In the function |x + 2|
, how is the graph altered?
In the function |x + 2|
, how is the graph altered?
Why are transformations useful in graphing functions?
Why are transformations useful in graphing functions?
Signup and view all the answers
What happens when a constant is subtracted inside a function's expression?
What happens when a constant is subtracted inside a function's expression?
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 parabolax^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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.