How to Use Desmos for Vertical Stretching of Functions

Understanding vertical stretching of functions is crucial in algebra, particularly when working with transformations of parent functions. In this guide, we'll explore how to use the Desmos graphing calculator to visualize and understand these transformations effectively.

### What is Vertical Stretching?
Vertical stretching occurs when a function is multiplied by a factor greater than 1. This transformation affects the height of the graph without changing its x-coordinates. For instance, if you have a parent function like \( f(x) = |x + 4| \), multiplying it by 3 gives you \( y = 3f(x) \) or \( y = 3|x + 4| \). This operation stretches the graph vertically, making it narrower and taller while keeping the vertex in the same place.

### Step-by-Step Guide to Using Desmos
1. **Graph the Parent Function**:
Open Desmos and enter the equation \( y = |x + 4| \). This represents the basic V-shaped graph. Notice its vertex is located at (-4, 0).

2. **Graph the Stretched Function**:
Next, input \( y = 3|x + 4| \) on a new line. You will see that this graph is steeper compared to the original. The vertex remains at (-4, 0), but the arms of the V are now further apart at any given x-value, indicating that the function's values have increased by a factor of 3.

3. **Using the Table Feature**:
To analyze specific points, utilize the table feature in Desmos. Click on the '+' icon and select 'table'. Input corresponding x-values, such as -6, -4, and -2, to see how the y-values change for both functions. For example, at x = -4, both functions yield a y-value of 0. However, at x = -2, the original function gives a y-value of 2, while the stretched function gives a y-value of 6, demonstrating the vertical stretching.

### Real-World Applications
Understanding vertical stretching is not just a theoretical exercise; it has real-world applications. For example, in physics, when analyzing forces or waves, the concept of stretching can help in understanding amplitude changes. Additionally, in economics, stretching can represent shifts in demand curves when prices increase.

### Conclusion
Using Desmos to visualize vertical stretching makes it easier to grasp the concept and apply it to various mathematical problems. By understanding how to manipulate functions and see their transformations, students can enhance their skills in algebra and related fields. Remember, practice with different functions and stretching factors to solidify your understanding!