The Function with Horizontal Stretching

Oct 13, 2025 by miodeya.com 113 views

Introduction

In mathematics, function transformations are a crucial concept in understanding how functions change when subjected to various transformations. One of the most common transformations is horizontal stretching, where the parent function is stretched horizontally by a factor of k. In this article, we will explore the effects of horizontal stretching on the function f(x) = -12x^2 + 21x - 3, and complete the table to show how the function has changed.

Horizontal Stretching

Horizontal stretching is a transformation that stretches the parent function horizontally by a factor of k. This means that the x-coordinates of the function are multiplied by k, resulting in a new function that is stretched horizontally. The general formula for horizontal stretching is:

g(x) = f(kx)

where g(x) is the new function, f(x) is the parent function, and k is the factor of stretching.

The Parent Function

The parent function is given by f(x) = -12x^2 + 21x - 3. This is a quadratic function that opens downwards, with a vertex at (h, k) = (-21/2(12), f(-21/2(12))). The function has a minimum value of f(-21/2(12)) = -3, and a maximum value of f(0) = -3.

The Function with Horizontal Stretching

To find the function with horizontal stretching, we need to substitute kx for x in the parent function. In this case, k = 1/2, so we substitute (1/2)x for x in the parent function:

g(x) = f((1/2)x) = -12((1/2)x)^2 + 21((1/2)x) - 3 = -12(1/4)x^2 + 21/2x - 3 = -3x^2 + 21/2x - 3

Table 1: The Function with Horizontal Stretching

x	f(x) = -12x^2 + 21x - 3	g(x) = -3x^2 + 21/2x - 3
-2	-3(-2)^2 + 21(-2) - 3	-3(-2)^2 + 21/2(-2) - 3
-1	-3(-1)^2 + 21(-1) - 3	-3(-1)^2 + 21/2(-1) - 3
0	-3(0)^2 + 21(0) - 3	-3(0)^2 + 21/2(0) - 3
1	-3(1)^2 + 21(1) - 3	-3(1)^2 + 21/2(1) - 3
2	-3(2)^2 + 21(2) - 3	-3(2)^2 + 21/2(2) - 3

Discussion

From the table, we can see that the function g(x) = -3x^2 + 21/2x - 3 is a horizontal stretch of the parent function f(x) = -12x^2 + 21x - 3 by a factor of k = 1/2. The function g(x) has a minimum value of g(-21/2(3)) = -3, and a maximum value of g(0) = -3.

Conclusion

In conclusion, horizontal stretching is a transformation that stretches the parent function horizontally by a factor of k. The function g(x) = -3x^2 + 21/2x - 3 is a horizontal stretch of the parent function f(x) = -12x^2 + 21x - 3 by a factor of k = 1/2. The function g(x) has a minimum value of g(-21/2(3)) = -3, and a maximum value of g(0) = -3.

References

[1] "Function Transformations" by Khan Academy
[2] "Horizontal Stretching" by Math Open Reference

Q: What is horizontal stretching in mathematics?

A: Horizontal stretching is a transformation that stretches the parent function horizontally by a factor of k. This means that the x-coordinates of the function are multiplied by k, resulting in a new function that is stretched horizontally.

Q: How do I apply horizontal stretching to a function?

A: To apply horizontal stretching to a function, you need to substitute kx for x in the parent function. For example, if the parent function is f(x) = -12x^2 + 21x - 3 and the factor of stretching is k = 1/2, you would substitute (1/2)x for x in the parent function to get g(x) = -3x^2 + 21/2x - 3.

Q: What is the effect of horizontal stretching on the graph of a function?

A: Horizontal stretching stretches the graph of a function horizontally by a factor of k. This means that the x-coordinates of the graph are multiplied by k, resulting in a new graph that is stretched horizontally.

Q: How does horizontal stretching affect the minimum and maximum values of a function?

A: Horizontal stretching does not affect the minimum and maximum values of a function. The minimum and maximum values of a function are determined by the vertex of the function, and horizontal stretching does not change the vertex.

Q: Can horizontal stretching be combined with other transformations?

A: Yes, horizontal stretching can be combined with other transformations, such as vertical stretching, horizontal shifting, and vertical shifting. However, the order in which you apply the transformations is important, as it can affect the final result.

Q: How do I determine the factor of stretching for a given function?

A: The factor of stretching is usually given in the problem, but if it is not given, you can determine it by looking at the graph of the function. The factor of stretching is the ratio of the x-coordinates of the graph to the x-coordinates of the parent function.

Q: What are some common applications of horizontal stretching in mathematics?

A: Horizontal stretching is used in a variety of mathematical applications, including:

Modeling real-world phenomena, such as population growth and decay
Solving optimization problems, such as finding the maximum or minimum value of a function
Analyzing the behavior of functions, such as finding the limits and derivatives of functions

Q: How can I practice applying horizontal stretching to functions?

A: You can practice applying horizontal stretching to functions by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and exercises to practice applying horizontal stretching.

Q: What are some common mistakes to avoid when applying horizontal stretching?

A: Some common mistakes to avoid when applying horizontal stretching include:

Forgetting to substitute kx for x in the parent function
Not considering the order in which you apply the transformations
Not checking the units of the function to ensure that they are consistent

Q: How can I use technology to help me apply horizontal stretching to functions?

A: You can use technology, such as graphing calculators or computer algebra systems, to help you apply horizontal stretching to functions. These tools can help you visualize the graph of the function and apply the transformations more easily.

Q: What are some resources for learning more about horizontal stretching?

A: Some resources for learning more about horizontal stretching include:

Textbooks and online resources, such as Khan Academy and Math Open Reference
Video tutorials and online courses, such as 3Blue1Brown and Crash Course
Practice problems and exercises, such as those found in your textbook or online resources

Conclusion

Horizontal stretching is an important concept in mathematics that can be used to model real-world phenomena and solve optimization problems. By understanding how to apply horizontal stretching to functions, you can gain a deeper understanding of the behavior of functions and improve your problem-solving skills.