4.4 4 Practice Modeling Stretching And Compressing Functions Answers

Alright, let's dive into understanding the concepts of stretching and compressing functions, particularly in the context of the 4.4.4 practice. This topic is crucial in understanding transformations of functions, and mastering it unlocks a deeper understanding of mathematical modeling.

Understanding Stretching and Compressing Functions

Function transformations allow us to manipulate the graph of a function by shifting, reflecting, stretching, or compressing it. Stretching and compressing functions involve altering the "shape" of the graph either vertically or horizontally. This is achieved by multiplying the function or the independent variable by a constant factor.

Vertical Stretching and Compressing

Vertical transformations affect the y-values of a function.

• Vertical Stretch: A vertical stretch occurs when we multiply the function f(x) by a constant a, where a > 1. The resulting function is g(x) = af(x)*. This transformation stretches the graph vertically away from the x-axis. Every y-value of f(x) is multiplied by a.
• Vertical Compression: A vertical compression occurs when we multiply the function f(x) by a constant a, where 0 < a < 1. The resulting function is g(x) = af(x)*. This transformation compresses the graph vertically towards the x-axis. Every y-value of f(x) is multiplied by a.

Horizontal Stretching and Compressing

Horizontal transformations affect the x-values of a function, and they work in a slightly counterintuitive way.

• Horizontal Compression: A horizontal compression occurs when we replace x with bx in the function f(x), where b > 1. The resulting function is g(x) = f(bx). This transformation compresses the graph horizontally towards the y-axis. To find the new x-value, divide the original x-value by b.
• Horizontal Stretch: A horizontal stretch occurs when we replace x with bx in the function f(x), where 0 < b < 1. The resulting function is g(x) = f(bx). This transformation stretches the graph horizontally away from the y-axis. To find the new x-value, divide the original x-value by b.

Key Differences

The key difference between vertical and horizontal stretches/compressions lies in where the constant is applied:

• Vertical: Constant is multiplied outside the function, affecting the y-values.
• Horizontal: Constant is multiplied inside the function (affecting the x-values), and the effect is inverse to what you might expect.

Practical Examples and Solutions (4.4.4 Practice)

Now, let's tackle some practical examples similar to what you might encounter in a 4.4.4 practice. We will go through examples that require determining the transformations applied to a function, as well as applying transformations to a function algebraically.

Example 1: Identifying Vertical Stretch

Suppose we have the function f(x) = x². We are given a transformed function g(x) = 3x². Describe the transformation.

Solution:

• Notice that g(x) = 3f(x).
• Since we are multiplying the entire function f(x) by 3, and 3 > 1, this is a vertical stretch by a factor of 3. This means every y-value of the original function is multiplied by 3.

Example 2: Identifying Vertical Compression

Consider f(x) = |x|. The transformed function is g(x) = (1/2)|x|. Describe the transformation.

Solution:

• Here, g(x) = (1/2)f(x).
• We are multiplying the function f(x) by 1/2, which is between 0 and 1. This indicates a vertical compression by a factor of 1/2. The graph is compressed towards the x-axis, and each y-value is halved.

Example 3: Identifying Horizontal Compression

Let f(x) = √x. We have a transformed function g(x) = √(4x). Describe the transformation.

Solution:

• We can rewrite g(x) as g(x) = f(4x).
• We are replacing x with 4x. Since 4 > 1, this represents a horizontal compression by a factor of 1/4. This means the graph is compressed horizontally towards the y-axis. To get the same y-value as the original function, the x-value must be 1/4 of the original.

Example 4: Identifying Horizontal Stretch

Suppose f(x) = sin(x). We have a transformed function g(x) = sin((1/3)x). Describe the transformation.

Solution:

• Here, g(x) = f((1/3)x).
• We are replacing x with (1/3)x. Since 1/3 is between 0 and 1, this indicates a horizontal stretch by a factor of 3. The graph is stretched horizontally away from the y-axis. To achieve the same y-value, the x-value must be three times the original.

Example 5: Applying Multiple Transformations

Let f(x) = x³. Find the equation of the function g(x) that is obtained by vertically stretching f(x) by a factor of 2 and then horizontally compressing it by a factor of 3.

Solution:

• Vertical Stretch by 2: This means we multiply the function by 2: 2f(x) = 2x³
• Horizontal Compression by 3: This means we replace x with 3x: 2f(3x) = 2(3x)³ = 2(27x³) = 54x³
• Therefore, g(x) = 54x³.

Example 6: Writing the Equation from a Description

The graph of y = |x| is vertically compressed by a factor of 1/4 and then horizontally stretched by a factor of 5. Write the equation of the transformed graph.

Solution:

• Vertical Compression by 1/4: This gives us (1/4)|x|.
• Horizontal Stretch by 5: This means we replace x with (1/5)x. So we have (1/4)|(1/5)x|.
• Therefore, the equation of the transformed graph is y = (1/4)|(1/5)x|.

Example 7: Determining Transformations from Two Points

Suppose f(x) passes through the point (2, 4). g(x) is a transformation of f(x) and passes through the point (6, 2). Describe the transformations that could have occurred.

Solution:

This example is more open-ended, and there are several possibilities. Let's analyze the changes:

• x-value: The x-value changed from 2 to 6, which could be a horizontal stretch. To get from 2 to 6, we multiply by 3. Therefore, one possibility is a horizontal stretch by a factor of 3, meaning we replaced x with (1/3)x.
• y-value: The y-value changed from 4 to 2, which could be a vertical compression. To get from 4 to 2, we multiply by 1/2. Therefore, another possibility is a vertical compression by a factor of 1/2, meaning we multiplied the function by 1/2.

One Possible Transformation: A horizontal stretch by a factor of 3 and a vertical compression by a factor of 1/2. In this case, g(x) = (1/2)f((1/3)x).

Example 8: Working with Tables of Values

Suppose you have a table of values for f(x):

Now, suppose g(x) = 2f(x) and h(x) = f(2x). Create tables of values for g(x) and h(x).

Solution:

For g(x) = 2f(x) (Vertical Stretch):

We multiply each f(x) value by 2. The x-values remain the same.

For h(x) = f(2x) (Horizontal Compression):

This is trickier. To get the x-values for h(x), we need to divide the original x-values by 2 (the inverse of the transformation). We need to find x values such that 2x matches the original x values.

For example:

• To find h(x) when x = -1, we need to find the f(x) value when x = 2(-1) = -2*. From the original table, f(-2) = 1. Therefore, h(-1) = 1.
• To find h(x) when x = 0, we need to find the f(x) value when x = 2(0) = 0*. From the original table, f(0) = 1. Therefore, h(0) = 1.
• To find h(x) when x = 1, we need to find the f(x) value when x = 2(1) = 2*. From the original table, f(2) = 1. Therefore, h(1) = 1.

We can add additional points by working backward. What if we want to know h(x) when x = -0.5? Then 2x = -1, and f(-1) = 0. So h(-0.5) = 0. Similarly, h(0.5) = 2 (because f(1) = 2).

Notice how the h(x) graph is compressed horizontally compared to the f(x) graph.

General Tips for Solving Stretching and Compressing Problems

1. Identify the Parent Function: Recognize the basic function that is being transformed (e.g., x², |x|, sin(x), √x).
2. Look for Multiplicative Constants: Pay attention to any numbers multiplying the function or the x-value.
3. Vertical vs. Horizontal: Remember that vertical transformations affect the y-values and are applied outside the function. Horizontal transformations affect the x-values and are applied inside the function, and their effect is the opposite of what you might intuitively expect.
4. Stretching vs. Compressing:
   • If the constant a is greater than 1, it's a stretch.
   • If the constant a is between 0 and 1, it's a compression.
5. Order of Transformations: If multiple transformations are applied, follow the order of operations (PEMDAS/BODMAS) in reverse. Horizontal shifts and stretches/compressions are generally applied before vertical shifts and stretches/compressions.
6. Use Test Points: If you're unsure, pick a few points on the original function's graph and see how they transform under the given transformations. This can help you visualize the changes and confirm your understanding.
7. Practice, Practice, Practice: The more you practice these types of problems, the more comfortable you will become with identifying and applying stretches and compressions.

Mathematical Justification

Why does multiplying x by a constant inside the function cause a horizontal stretch or compression, and why is the effect seemingly "reversed?"

Let's consider f(x) and g(x) = f(bx). We want to understand how the x-values in g(x) relate to the x-values in f(x) to produce the same y-value.

Suppose we want g(x) to have the same y-value as f(x₀), where x₀ is a specific x-value. That is, we want g(x) = f(x₀).

Since g(x) = f(bx), we have f(bx) = f(x₀). For this to be true, we must have bx = x₀. Solving for x, we get x = x₀/b.

This equation, x = x₀/b, is the key. It tells us that to get the same y-value in g(x) as we had in f(x) at x₀, we need to use an x-value that is x₀ divided by b.

• If b > 1: Then x = x₀/b is smaller than x₀. This means the graph of g(x) is compressed horizontally towards the y-axis.
• If 0 < b < 1: Then x = x₀/b is larger than x₀. This means the graph of g(x) is stretched horizontally away from the y-axis.

In simpler terms, when b > 1, you need a smaller x-value in g(x) to achieve the same y-value as in f(x), hence the compression. When 0 < b < 1, you need a larger x-value in g(x) to achieve the same y-value as in f(x), hence the stretch.

Common Mistakes to Avoid

• Confusing Vertical and Horizontal Transformations: Always remember where the constant is being multiplied (inside or outside the function).
• Incorrectly Applying Horizontal Stretches/Compressions: Remember that the effect of the constant inside the function is inverse to what you might expect.
• Forgetting the Order of Operations: Apply transformations in the correct order.
• Not Understanding the Effect on Individual Points: If you're struggling, pick a few points on the original graph and track how they change under the transformations.
• Assuming the Same Rules Apply to All Functions: While the principles are the same, the visual effect of a transformation can differ depending on the shape of the parent function.

Conclusion

Mastering stretching and compressing functions is a fundamental skill in understanding function transformations. By understanding the underlying principles and working through various examples, you can confidently tackle problems related to this topic. Remember to pay close attention to whether the transformation is vertical or horizontal and whether it's a stretch or a compression. Regular practice and careful attention to detail will solidify your understanding and make you proficient in this area of mathematics. With a solid grasp of these concepts, you'll be well-equipped to handle more complex transformations and applications in mathematical modeling.