What Kind Of Transformation Converts The Graph Of F ( X ) = − 4 ( X − 6 ) 2 − 7 F(x)=-4(x-6)^2-7 F ( X ) = − 4 ( X − 6 ) 2 − 7 Into The Graph Of G ( X ) = − 4 ( X + 1 ) 2 − 4 G(x)=-4(x+1)^2-4 G ( X ) = − 4 ( X + 1 ) 2 − 4 ?A. Translation 7 Units Left And 3 Units DownB. Translation 7 Units Right And 3 Units UpC. Translation 7 Units Right And 3 Units

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What kind of transformation converts the graph of f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7 into the graph of g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4?

Understanding Graph Transformations

Graph transformations are essential in mathematics, particularly in algebra and calculus. They help us understand how functions change when certain operations are applied to them. In this article, we will explore the transformation that converts the graph of f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7 into the graph of g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4.

The Basics of Graph Transformations

Before we dive into the specific transformation, let's review the basics of graph transformations. There are four main types of transformations:

  1. Translation: This involves moving the graph of a function to a new position on the coordinate plane. It can be horizontal, vertical, or a combination of both.
  2. Dilation: This involves changing the size of the graph of a function. It can be a stretch or a shrink.
  3. Rotation: This involves rotating the graph of a function around a fixed point.
  4. Reflection: This involves flipping the graph of a function over a fixed line.

The Transformation in Question

Now, let's analyze the transformation that converts the graph of f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7 into the graph of g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4. To do this, we need to compare the two functions and identify the changes that have been made.

Comparing the Two Functions

The two functions are:

f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7

g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4

Let's compare the two functions:

  • The coefficient of the squared term is the same in both functions, which is -4.
  • The constant term is different in both functions, which is -7 in f(x)f(x) and -4 in g(x)g(x).
  • The term inside the squared term is different in both functions, which is (x−6)(x-6) in f(x)f(x) and (x+1)(x+1) in g(x)g(x).

Identifying the Transformation

Based on the comparison, we can see that the transformation involves a change in the term inside the squared term. Specifically, (x−6)(x-6) has been replaced by (x+1)(x+1). This indicates that the graph of f(x)f(x) has been translated to the right by 7 units and down by 3 units to obtain the graph of g(x)g(x).

Conclusion

In conclusion, the transformation that converts the graph of f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7 into the graph of g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4 is a translation of 7 units right and 3 units down.

The Correct Answer

The correct answer is:

A. Translation 7 units left and 3 units down

Note: The answer is actually the opposite of what we concluded. This is because the transformation involves a change in the term inside the squared term, which indicates a translation to the right, not left.

Why the Mistake?

The mistake occurred because we incorrectly identified the transformation. We assumed that the change in the term inside the squared term indicated a translation to the left, when in fact it indicated a translation to the right.

The Correct Transformation

The correct transformation is:

A. Translation 7 units right and 3 units down

The Correct Reasoning

The correct reasoning is as follows:

  • The term inside the squared term has changed from (x−6)(x-6) to (x+1)(x+1), which indicates a translation to the right by 7 units.
  • The constant term has changed from -7 to -4, which indicates a translation down by 3 units.

Conclusion

In conclusion, the transformation that converts the graph of f(x)=−4(x−6)2−7f(x)=-4(x-6)^2-7 into the graph of g(x)=−4(x+1)2−4g(x)=-4(x+1)^2-4 is a translation of 7 units right and 3 units down.

The Final Answer

The final answer is:

A. Translation 7 units right and 3 units down
Q&A: Graph Transformations

Understanding Graph Transformations

Graph transformations are essential in mathematics, particularly in algebra and calculus. They help us understand how functions change when certain operations are applied to them. In this article, we will explore some common questions and answers related to graph transformations.

Q: What is a graph transformation?

A: A graph transformation is a change in the graph of a function that results from applying certain operations to the function. These operations can include translations, dilations, rotations, and reflections.

Q: What are the four main types of graph transformations?

A: The four main types of graph transformations are:

  1. Translation: This involves moving the graph of a function to a new position on the coordinate plane. It can be horizontal, vertical, or a combination of both.
  2. Dilation: This involves changing the size of the graph of a function. It can be a stretch or a shrink.
  3. Rotation: This involves rotating the graph of a function around a fixed point.
  4. Reflection: This involves flipping the graph of a function over a fixed line.

Q: What is a translation in graph transformations?

A: A translation in graph transformations involves moving the graph of a function to a new position on the coordinate plane. It can be horizontal, vertical, or a combination of both. For example, if a function is translated 3 units to the right, the graph of the function will be shifted 3 units to the right.

Q: What is a dilation in graph transformations?

A: A dilation in graph transformations involves changing the size of the graph of a function. It can be a stretch or a shrink. For example, if a function is dilated by a factor of 2, the graph of the function will be stretched by a factor of 2.

Q: What is a rotation in graph transformations?

A: A rotation in graph transformations involves rotating the graph of a function around a fixed point. For example, if a function is rotated 90 degrees counterclockwise, the graph of the function will be rotated 90 degrees counterclockwise.

Q: What is a reflection in graph transformations?

A: A reflection in graph transformations involves flipping the graph of a function over a fixed line. For example, if a function is reflected over the x-axis, the graph of the function will be flipped over the x-axis.

Q: How do I determine the type of graph transformation that has occurred?

A: To determine the type of graph transformation that has occurred, you need to analyze the changes that have been made to the function. Look for changes in the term inside the squared term, the coefficient of the squared term, and the constant term. These changes can indicate a translation, dilation, rotation, or reflection.

Q: Can you give an example of a graph transformation?

A: Yes, here is an example of a graph transformation:

Suppose we have a function f(x) = x^2 + 3x - 4. If we translate this function 2 units to the right, the new function will be f(x) = (x - 2)^2 + 3(x - 2) - 4.

Q: How do I graph a function after a graph transformation?

A: To graph a function after a graph transformation, you need to apply the transformation to the original graph. For example, if a function is translated 2 units to the right, you need to shift the original graph 2 units to the right.

Conclusion

In conclusion, graph transformations are essential in mathematics, particularly in algebra and calculus. They help us understand how functions change when certain operations are applied to them. By understanding the four main types of graph transformations and how to determine the type of transformation that has occurred, you can graph functions after a graph transformation.

The Final Answer

The final answer is:

  • A translation involves moving the graph of a function to a new position on the coordinate plane.
  • A dilation involves changing the size of the graph of a function.
  • A rotation involves rotating the graph of a function around a fixed point.
  • A reflection involves flipping the graph of a function over a fixed line.

Common Graph Transformations

Here are some common graph transformations:

  • Translation: f(x) = (x - h)^2 + k
  • Dilation: f(x) = a(x - h)^2 + k
  • Rotation: f(x) = (x - h)^2 + k
  • Reflection: f(x) = -(x - h)^2 + k

Graph Transformation Practice

Here are some practice problems to help you understand graph transformations:

  1. Translate the function f(x) = x^2 + 3x - 4 2 units to the right.
  2. Dilate the function f(x) = x^2 + 3x - 4 by a factor of 2.
  3. Rotate the function f(x) = x^2 + 3x - 4 90 degrees counterclockwise.
  4. Reflect the function f(x) = x^2 + 3x - 4 over the x-axis.

Answer Key

  1. f(x) = (x - 2)^2 + 3(x - 2) - 4
  2. f(x) = 2(x^2 + 3x - 4)
  3. f(x) = -(x^2 + 3x - 4)
  4. f(x) = -(x^2 + 3x - 4)