Vertical stretch and horizontal stretch are two fundamental transformations in mathematics that manipulate the shape of graphs. Vertical stretch affects the distance from the x-axis for any given value of x, while horizontal stretch modifies the distance between points on the graph for any given value of y. Together, these transformations can be used to create various effects, such as distorting the shape of a function or shifting its position on the coordinate plane.
Stretching Functions Vertically and Horizontally
When transforming functions, we often use vertical and horizontal stretching to alter their graphs. Understanding the structure of these stretches is critical for accurate transformations.
Vertical Stretch
To vertically stretch a function by a factor of “a”, we multiply the function by “a”. This:
- Stretches the graph vertically by a factor of “a” if a > 1.
- Compresses the graph vertically by a factor of “a” if 0 < a < 1.
Structure of a Vertical Stretch:
y = a * f(x)
- “a” is the stretching factor.
- “f(x)” is the original function.
Horizontal Stretch
To horizontally stretch a function by a factor of “a”, we replace “x” with “x/a” in the function. This:
- Stretches the graph horizontally by a factor of “a” if a > 1.
- Compresses the graph horizontally by a factor of “a” if 0 < a < 1.
Structure of a Horizontal Stretch:
y = f(x/a)
- “a” is the stretching factor.
- “f(x)” is the original function.
Combined Stretching
We can combine both vertical and horizontal stretching by applying the transformations in any order. The result is a function that is:
Structure of Combined Stretching:
| Transformation | Result |
|---|---|
| Vertical stretch by “a” followed by horizontal stretch by “b” | y = a * f(b * x) |
| Horizontal stretch by “b” followed by vertical stretch by “a” | y = a * f(x/b) |
Example
To vertically stretch the function y = x^2 by a factor of 3 and horizontally stretch it by a factor of 2, we would:
- Vertical stretch: y = 3 * x^2
- Horizontal stretch: y = 3 * (x/2)^2
- Final function: y = 3(x^2)/4
Question 1:
What is the difference between a vertical stretch and a horizontal stretch?
Answer:
- A vertical stretch changes the range of the function, multiplying the output values by a constant.
- A horizontal stretch changes the domain of the function, adding a constant to the input values.
Question 2:
How does a vertical stretch affect the graph of a function?
Answer:
- A vertical stretch by a factor of “a” makes the graph taller or shorter by a factor of “a”.
- Positive stretches make the graph taller, and negative stretches make it shorter.
Question 3:
What is the effect of a horizontal stretch on the graph of a function?
Answer:
- A horizontal stretch by a factor of “a” makes the graph wider or narrower by a factor of “a”.
- Positive stretches make the graph wider, and negative stretches make it narrower.
That’s it for our dive into vertical and horizontal stretches! We hope you found this helpful and if you’re ever feeling a bit rusty, just swing by again. Remember, practice makes perfect, so keep stretching those graphs and you’ll be a pro in no time. Thanks for reading, and see you next time!