Consider The Function $f(x) = 2x - 6$.Match Each Transformation Of $f(x$\] With Its Description. Not All Tiles Will Be Used.- Shifts $f(x$\] 4 Units Down: $\square$- Compresses $f(x$\] By A Factor Of
Introduction
In mathematics, transformations of linear functions are essential concepts that help us understand how functions can be manipulated and represented in different ways. A linear function is a polynomial function of degree one, and it can be represented in the form of f(x) = mx + b, where m is the slope and b is the y-intercept. In this article, we will consider the function f(x) = 2x - 6 and match each transformation of f(x) with its description.
Understanding the Original Function
The original function is f(x) = 2x - 6. This function has a slope of 2 and a y-intercept of -6. The graph of this function is a straight line with a positive slope, indicating that it increases as x increases.
Shifting the Function
Shifting a function means moving it up or down by a certain number of units. To shift a function down by a certain number of units, we need to subtract that number from the function.
Shifts f(x) 4 units down
To shift f(x) = 2x - 6 4 units down, we need to subtract 4 from the function. This can be represented as:
f(x) = 2x - 6 - 4 f(x) = 2x - 10
This new function has the same slope as the original function but is shifted down by 4 units.
Shifts f(x) 4 units up
To shift f(x) = 2x - 6 4 units up, we need to add 4 to the function. This can be represented as:
f(x) = 2x - 6 + 4 f(x) = 2x - 2
This new function has the same slope as the original function but is shifted up by 4 units.
Compressing the Function
Compressing a function means making it narrower or wider. To compress a function by a certain factor, we need to multiply the function by that factor.
Compresses f(x) by a factor of 1/2
To compress f(x) = 2x - 6 by a factor of 1/2, we need to multiply the function by 1/2. This can be represented as:
f(x) = (1/2)(2x - 6) f(x) = x - 3
This new function has a slope of 1, which is half of the original slope, indicating that it is compressed by a factor of 1/2.
Compresses f(x) by a factor of 2
To compress f(x) = 2x - 6 by a factor of 2, we need to multiply the function by 2. This can be represented as:
f(x) = 2(2x - 6) f(x) = 4x - 12
This new function has a slope of 4, which is twice the original slope, indicating that it is compressed by a factor of 2.
Stretching the Function
Stretching a function means making it wider or narrower. To stretch a function by a certain factor, we need to multiply the function by that factor.
Stretches f(x) by a factor of 2
To stretch f(x) = 2x - 6 by a factor of 2, we need to multiply the function by 2. This can be represented as:
f(x) = 2(2x - 6) f(x) = 4x - 12
This new function has a slope of 4, which is twice the original slope, indicating that it is stretched by a factor of 2.
Stretches f(x) by a factor of 1/2
To stretch f(x) = 2x - 6 by a factor of 1/2, we need to multiply the function by 1/2. This can be represented as:
f(x) = (1/2)(2x - 6) f(x) = x - 3
This new function has a slope of 1, which is half of the original slope, indicating that it is stretched by a factor of 1/2.
Reflecting the Function
Reflecting a function means flipping it over a certain line. To reflect a function over the x-axis, we need to multiply the function by -1.
Reflects f(x) over the x-axis
To reflect f(x) = 2x - 6 over the x-axis, we need to multiply the function by -1. This can be represented as:
f(x) = -1(2x - 6) f(x) = -2x + 6
This new function has a slope of -2, which is the negative of the original slope, indicating that it is reflected over the x-axis.
Reflects f(x) over the y-axis
To reflect f(x) = 2x - 6 over the y-axis, we need to replace x with -x in the function. This can be represented as:
f(x) = 2(-x) - 6 f(x) = -2x - 6
This new function has a slope of -2, which is the negative of the original slope, indicating that it is reflected over the y-axis.
Conclusion
Introduction
In our previous article, we discussed the transformations of linear functions, including shifting, compressing, stretching, and reflecting. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on these transformations.
Q: What is the difference between compressing and stretching a function?
A: Compressing a function means making it narrower or wider, while stretching a function means making it wider or narrower. In other words, compressing a function reduces its slope, while stretching a function increases its slope.
Q: How do I compress a function by a factor of 1/2?
A: To compress a function by a factor of 1/2, you need to multiply the function by 1/2. For example, if you have the function f(x) = 2x - 6, you can compress it by a factor of 1/2 by multiplying it by 1/2, resulting in f(x) = (1/2)(2x - 6) = x - 3.
Q: How do I stretch a function by a factor of 2?
A: To stretch a function by a factor of 2, you need to multiply the function by 2. For example, if you have the function f(x) = 2x - 6, you can stretch it by a factor of 2 by multiplying it by 2, resulting in f(x) = 2(2x - 6) = 4x - 12.
Q: What is the effect of reflecting a function over the x-axis?
A: Reflecting a function over the x-axis means flipping it over the x-axis. This results in a new function with a slope that is the negative of the original slope. For example, if you have the function f(x) = 2x - 6, reflecting it over the x-axis results in f(x) = -2x + 6.
Q: What is the effect of reflecting a function over the y-axis?
A: Reflecting a function over the y-axis means replacing x with -x in the function. This results in a new function with a slope that is the negative of the original slope. For example, if you have the function f(x) = 2x - 6, reflecting it over the y-axis results in f(x) = -2x - 6.
Q: Can I combine multiple transformations to create a new function?
A: Yes, you can combine multiple transformations to create a new function. For example, you can compress a function by a factor of 1/2 and then reflect it over the x-axis. The order in which you apply the transformations does not matter, as long as you apply them correctly.
Q: How do I determine the correct order of transformations?
A: To determine the correct order of transformations, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate any expressions inside parentheses first.
- Exponents: Evaluate any exponents next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following this order of operations, you can ensure that you apply the transformations in the correct order.
Conclusion
In this Q&A article, we have provided answers to common questions about transformations of linear functions. We have discussed compressing, stretching, reflecting, and combining multiple transformations, and we have provided examples to help illustrate each concept. By understanding these transformations, you can better analyze and represent linear functions in different ways.