Transformations

Definition

Transformations refer to the various ways in which the graph of a function can be altered or modified, such as shifting, stretching, compressing, or reflecting the original function. These changes to the function's graph can significantly impact the function's behavior and properties, making transformations an essential concept in understanding and working with various types of functions, including absolute value functions.

5 Must Know Facts For Your Next Test

1. Transformations can be applied to absolute value functions to create new functions with different properties and behaviors.
2. The general form of an absolute value function with transformations is $f(x) = a|b(x - h)| + k$, where $a$, $b$, $h$, and $k$ are real numbers that control the transformations.
3. Transformations can affect the domain, range, intercepts, and symmetry of an absolute value function.
4. Translations of an absolute value function shift the graph horizontally and/or vertically, without changing its shape.
5. Dilations of an absolute value function stretch or compress the graph along the horizontal and/or vertical axis, changing the steepness of the function.

Review Questions

• Explain how the parameters $a$, $b$, $h$, and $k$ in the general form of an absolute value function with transformations affect the graph of the function.
  • The parameter $a$ controls the vertical stretch or compression of the graph, with $a > 1$ causing a vertical stretch and $0 < a < 1$ causing a vertical compression. The parameter $b$ controls the horizontal stretch or compression, with $b > 1$ causing a horizontal stretch and $0 < b < 1$ causing a horizontal compression. The parameter $h$ controls the horizontal shift of the graph, with $h > 0$ causing a shift to the right and $h < 0$ causing a shift to the left. The parameter $k$ controls the vertical shift of the graph, with $k > 0$ causing an upward shift and $k < 0$ causing a downward shift.
• Describe how transformations can affect the domain and range of an absolute value function.
  • Transformations can alter the domain and range of an absolute value function. Horizontal shifts (changes to $h$) affect the domain, as they move the function left or right on the x-axis. Vertical shifts (changes to $k$) affect the range, as they move the function up or down on the y-axis. Dilations (changes to $a$ and $b$) can also impact the domain and range by stretching or compressing the function along the horizontal and/or vertical axes. Understanding how these transformations affect the domain and range is crucial for accurately graphing and working with absolute value functions.
• Analyze how transformations can be used to model real-world situations involving absolute value functions.
  • Transformations of absolute value functions can be used to model a variety of real-world scenarios. For example, the absolute value function can represent the distance between two points, and transformations can be used to model changes in this distance based on factors such as location, time, or other variables. Transformations can also be used to model financial situations, such as the absolute value of profit or loss, or engineering problems, such as the absolute value of displacement or velocity. By understanding how to apply transformations to absolute value functions, students can develop the skills to analyze and solve complex real-world problems involving these types of functions.

Related terms

Translation: A transformation that shifts the graph of a function horizontally or vertically without changing its shape or size.

Dilation: A transformation that scales the graph of a function, either stretching or compressing it along the horizontal or vertical axis.

Reflection:

A transformation that flips the graph of a function over a line, either the x-axis or the y-axis.