Translation

Definition

Translation refers to the process of shifting a graph of a function in a specific direction without changing its shape or size. This movement is defined by adding or subtracting constants from the function's variables, resulting in a new graph that retains the same characteristics but is positioned differently in the coordinate plane.

5 Must Know Facts For Your Next Test

1. A translation can be horizontal, vertical, or both, depending on whether constants are added or subtracted from the x-coordinate, y-coordinate, or both in the function's equation.
2. For a horizontal translation, if you add a constant 'h' to 'x', the graph shifts left if 'h' is negative and right if 'h' is positive.
3. For a vertical translation, adding a constant 'k' to 'f(x)' shifts the graph upward if 'k' is positive and downward if 'k' is negative.
4. Translations do not affect the domain or range of the function; they simply change the location of the graph within the same coordinate system.
5. In function notation, if f(x) represents the original function, then f(x - h) + k represents the translated function, indicating a shift by 'h' units horizontally and 'k' units vertically.

Review Questions

• How does adding or subtracting constants from the variables of a function affect its graph?
  • Adding or subtracting constants from the variables in a function results in a translation of its graph. Specifically, modifying the x-coordinate with a constant shifts the graph horizontally while adjusting the y-coordinate with a constant shifts it vertically. This means that translations change where the graph appears on the coordinate plane but do not alter its shape or size.
• Compare and contrast translation with reflection and scaling transformations in terms of their impact on graph behavior.
  • Translation differs from reflection and scaling as it maintains both the shape and size of a graph while shifting its position. In contrast, reflection creates a mirror image over an axis, altering orientation without changing dimensions, while scaling modifies the size and shape by stretching or compressing it. All three transformations are crucial for understanding how functions behave on graphs, but they achieve different effects.
• Evaluate how understanding translations aids in solving complex problems involving composite functions and transformations.
  • Understanding translations is key when working with composite functions because it allows students to visualize how one function's output can shift another function's input. This insight can simplify problem-solving by making it easier to determine how transformations affect combinations of functions. Additionally, mastering translations helps in identifying patterns within complex problems, ultimately enhancing analytical skills and mathematical reasoning.

Related terms

Reflection:

Reflection is a transformation that flips a graph over a specific axis, creating a mirror image of the original graph.

Scaling: Scaling is the process of resizing a graph by stretching or compressing it vertically or horizontally, which alters the shape and size of the original graph.

Function Notation: Function notation is a way to represent functions and their outputs, often written as f(x), which clearly identifies the relationship between inputs and outputs.