Graph transformations

Definition

Graph transformations refer to the various operations that can be performed on the graph of a function to change its position, shape, or orientation. These transformations include shifting, reflecting, stretching, and compressing the graph, which allows for a visual understanding of how alterations in the function's equation affect its graphical representation.

5 Must Know Facts For Your Next Test

1. Horizontal translations of the graph occur when you add or subtract a constant from the input variable, affecting its position along the x-axis.
2. Vertical translations happen when you add or subtract a constant from the output of the function, moving the graph up or down.
3. Reflections can be achieved by negating either the output (for reflection across the x-axis) or the input (for reflection across the y-axis) in the function's equation.
4. Stretches and compressions occur when you multiply the function by a constant; values greater than one will stretch the graph while values between zero and one will compress it.
5. Understanding these transformations helps in sketching graphs of complex functions by applying these operations step-by-step to simpler parent functions.

Review Questions

• How do horizontal and vertical translations affect the graph of a function, and what are their respective impacts on the function's equation?
  • Horizontal translations affect the graph by shifting it left or right depending on whether you add or subtract from the input variable. For instance, changing $$f(x)$$ to $$f(x - c)$$ shifts the graph right by $$c$$ units. Vertical translations shift the graph up or down by adding or subtracting from the function's output. For example, changing $$f(x)$$ to $$f(x) + c$$ moves it up by $$c$$ units.
• What role do reflections play in graph transformations, and how would you reflect the graph of a function across both axes?
  • Reflections change the orientation of a graph without altering its shape. To reflect a graph across the x-axis, you negate the output of the function, changing $$f(x)$$ to $$-f(x)$$. To reflect it across the y-axis, you negate the input, changing $$f(x)$$ to $$f(-x)$$. This allows for easy visualization of how negative values affect the overall appearance of a function.
• Evaluate how combining multiple transformations can be used to derive a new function's graph from its parent function and discuss any potential complexities that arise.
  • Combining multiple transformations involves sequentially applying each transformation to a parent function, which can lead to complex interactions between shifts, stretches, and reflections. For example, starting with a parent function like $$f(x) = x^2$$, if we first reflect it across the x-axis to get $$-f(x)$$ and then shift it up by 3 units to get $$-x^2 + 3$$, we end up with a completely different shape and position than just performing one transformation alone. This process emphasizes the need to carefully track each step to ensure accurate representation of each operation's cumulative effect.

Related terms

Translation:

A transformation that shifts a graph horizontally or vertically without changing its shape or orientation.

Reflection:

A transformation that flips a graph over a specific line, such as the x-axis or y-axis, altering its orientation.

Dilation: A transformation that stretches or compresses a graph either vertically or horizontally, affecting its size but not its shape.