Vertical dilation and horizontal dilation are mathematical transformations that alter the shape of a function. Vertical dilation, also known as vertical stretching or shrinking, affects the range or the y-coordinates of the function. Horizontal dilation, also known as horizontal stretching or shrinking, affects the domain or the x-coordinates of the function. By understanding these transformations, one can effectively manipulate functions to modify their characteristics and solve mathematical problems.
Effects of Vertical and Horizontal Dilation on Graphs
Vertical and horizontal dilations are transformations that alter the size and shape of a graph while maintaining its overall shape. Understanding their distinct effects is crucial for comprehending graph transformations.
Vertical Dilation
- Effect: Stretches or shrinks the graph vertically.
- Formula: y = k * f(x), where k is the constant that determines the dilation factor.
- k > 1: Stretches the graph vertically.
- 0 < k < 1: Shrinks the graph vertically.
- k = 0: Folds the graph onto the x-axis.
- k < 0: Flips the graph over the x-axis.
- Graph:
- Vertical stretch: Graph moves up for k > 1 and down for 0 < k < 1.
- Vertical shrink: Graph moves down for k > 1 and up for 0 < k < 1.
Horizontal Dilation
- Effect: Stretches or shrinks the graph horizontally.
- Formula: y = f(kx), where k is the constant that determines the dilation factor.
- k > 1: Shrinks the graph horizontally.
- 0 < k < 1: Stretches the graph horizontally.
- k = 0: Folds the graph onto the y-axis.
- k < 0: Flips the graph over the y-axis.
- Graph:
- Horizontal stretch: Graph moves left for k > 1 and right for 0 < k < 1.
- Horizontal shrink: Graph moves right for k > 1 and left for 0 < k < 1.
Table Summary
| Transformation | Formula | Effect |
|---|---|---|
| Vertical Dilation | y = k * f(x) | Stretches or shrinks vertically |
| Horizontal Dilation | y = f(kx) | Stretches or shrinks horizontally |
Example
Consider the graph of y = x^2.
- Vertical dilation with k = 2: y = 2 * x^2
- Stretches the graph vertically by a factor of 2.
- Horizontal dilation with k = 1/2: y = x^(2/1)
- Stretches the graph horizontally by a factor of 2.
Question 1:
What are the key differences between vertical and horizontal dilation?
Answer:
- Vertical dilation stretches or shrinks the graph of a function vertically, changing the range of values but leaving the domain unaltered.
- Horizontal dilation stretches or shrinks the graph of a function horizontally, changing the domain of values but leaving the range unaltered.
Question 2:
How does the coefficient of dilation affect the stretch or shrink?
Answer:
- A dilation factor of greater than 1 stretches the graph away from the origin, while a factor of less than 1 shrinks it towards the origin.
Question 3:
What is the effect of the point around which the dilation is performed?
Answer:
- The point around which the dilation is performed acts as the fixed point, remaining unchanged after the transformation.
Thanks for sticking with me through this journey of vertical versus horizontal dilation. I hope you now have a clearer understanding of how these transformations affect the graph of a function. If you still have questions, feel free to drop a comment below and I’ll do my best to help. Remember, practice makes perfect, so keep graphing those equations and you’ll become a dilation pro in no time. Until next time, keep exploring the wonderful world of mathematics!