The Effect of the Transformation on the Range of f

Oct 13, 2025 by windailys.com 146 views

Understanding the Original Function f

The original function f has a range of -6 ≤ y ≤ 4 in the xy-plane. This means that for every value of x, the corresponding value of y will fall within this range. In other words, the function f maps every input x to an output y that lies between -6 and 4.

The Transformation g(x) = -2f(x/3) + 8

The graph of y = g(x) is the result of a transformation of the original function f. The transformation is given by the equation g(x) = -2f(x/3) + 8. This equation involves three main components:

Horizontal Stretching: The function f is stretched horizontally by a factor of 3. This means that the x-coordinates of the graph of f are divided by 3 to obtain the x-coordinates of the graph of g.
Vertical Stretching: The function f is stretched vertically by a factor of 2. This means that the y-coordinates of the graph of f are multiplied by 2 to obtain the y-coordinates of the graph of g.
Vertical Translation: The function f is translated vertically by 8 units upwards. This means that the y-coordinates of the graph of f are increased by 8 to obtain the y-coordinates of the graph of g.
Reflection: The function f is reflected about the x-axis. This means that the y-coordinates of the graph of f are multiplied by -1 to obtain the y-coordinates of the graph of g.

The Effect of the Transformation on the Range of f

The transformation g(x) = -2f(x/3) + 8 affects the range of the original function f. Since the function f has a range of -6 ≤ y ≤ 4, the transformed function g will have a range of -2(-6) ≤ y ≤ 2(4) - 8.

Calculating the New Range

To calculate the new range, we need to evaluate the expressions -2(-6) and 2(4) - 8.

Lower Bound: The lower bound of the new range is -2(-6) = 12.
Upper Bound: The upper bound of the new range is 2(4) - 8 = 0.

The New Range of g

Therefore, the new range of the function g is 12 ≤ y ≤ 0.

Conclusion

In conclusion, the transformation g(x) = -2f(x/3) + 8 affects the range of the original function f. The new range of the function g is 12 ≤ y ≤ 0.

Key Takeaways

Horizontal Stretching: The function f is stretched horizontally by a factor of 3.
Vertical Stretching: The function f is stretched vertically by a factor of 2.
Vertical Translation: The function f is translated vertically by 8 units upwards.
Reflection: The function f is reflected about the x-axis.
New Range: The new range of the function g is 12 ≤ y ≤ 0.

Final Thoughts

The transformation g(x) = -2f(x/3) + 8 is a complex transformation that involves horizontal stretching, vertical stretching, vertical translation, and reflection. The new range of the function g is 12 ≤ y ≤ 0, which is different from the original range of the function f. This transformation can be used to create new functions with different ranges and properties.

Q1: What is the effect of the horizontal stretching on the graph of f?

A1: The horizontal stretching of the function f by a factor of 3 means that the x-coordinates of the graph of f are divided by 3 to obtain the x-coordinates of the graph of g. This results in a wider graph of g compared to the graph of f.

Q2: How does the vertical stretching affect the graph of f?

A2: The vertical stretching of the function f by a factor of 2 means that the y-coordinates of the graph of f are multiplied by 2 to obtain the y-coordinates of the graph of g. This results in a taller graph of g compared to the graph of f.

Q3: What is the effect of the vertical translation on the graph of f?

A3: The vertical translation of the function f by 8 units upwards means that the y-coordinates of the graph of f are increased by 8 to obtain the y-coordinates of the graph of g. This results in a graph of g that is shifted upwards compared to the graph of f.

Q4: How does the reflection about the x-axis affect the graph of f?

A4: The reflection of the function f about the x-axis means that the y-coordinates of the graph of f are multiplied by -1 to obtain the y-coordinates of the graph of g. This results in a graph of g that is a mirror image of the graph of f about the x-axis.

Q5: What is the new range of the function g?

A5: The new range of the function g is 12 ≤ y ≤ 0. This is different from the original range of the function f, which is -6 ≤ y ≤ 4.

Q6: How can the transformation g(x) = -2f(x/3) + 8 be used in real-world applications?

A6: The transformation g(x) = -2f(x/3) + 8 can be used in various real-world applications, such as:

Signal Processing: The transformation can be used to filter out noise from a signal and to amplify the signal.
Image Processing: The transformation can be used to enhance the contrast of an image and to remove noise from the image.
Data Analysis: The transformation can be used to analyze and visualize data, such as stock prices or weather patterns.

Q7: What are some common mistakes to avoid when applying the transformation g(x) = -2f(x/3) + 8?

A7: Some common mistakes to avoid when applying the transformation g(x) = -2f(x/3) + 8 include:

Incorrectly applying the horizontal stretching: Make sure to divide the x-coordinates of the graph of f by 3 to obtain the x-coordinates of the graph of g.
Incorrectly applying the vertical stretching: Make sure to multiply the y-coordinates of the graph of f by 2 to obtain the y-coordinates of the graph of g.
Incorrectly applying the vertical translation: Make sure to increase the y-coordinates of the graph of f by 8 to obtain the y-coordinates of the graph of g.
Incorrectly applying the reflection: Make sure to multiply the y-coordinates of the graph of f by -1 to obtain the y-coordinates of the graph of g.

Q8: How can the transformation g(x) = -2f(x/3) + 8 be generalized to other functions?

A8: The transformation g(x) = -2f(x/3) + 8 can be generalized to other functions by replacing the function f with the new function. For example, if we have a function h(x) and we want to apply the transformation g(x) = -2h(x/3) + 8, we can simply replace f(x) with h(x) in the original transformation.

Q9: What are some advanced topics related to the transformation g(x) = -2f(x/3) + 8?

A9: Some advanced topics related to the transformation g(x) = -2f(x/3) + 8 include:

Parametric Transformations: The transformation g(x) = -2f(x/3) + 8 can be generalized to parametric transformations, which involve multiple parameters.
Non-Linear Transformations: The transformation g(x) = -2f(x/3) + 8 can be generalized to non-linear transformations, which involve non-linear functions.
Fractal Geometry: The transformation g(x) = -2f(x/3) + 8 can be used to generate fractals, which are geometric shapes that exhibit self-similarity.

Q10: How can the transformation g(x) = -2f(x/3) + 8 be used in machine learning and artificial intelligence?

A10: The transformation g(x) = -2f(x/3) + 8 can be used in machine learning and artificial intelligence to:

Preprocess data: The transformation can be used to preprocess data before feeding it into a machine learning model.
Feature extraction: The transformation can be used to extract features from data, which can be used to train a machine learning model.
Data augmentation: The transformation can be used to augment data, which can be used to train a machine learning model.