Graph The Function $f(x) = (x+2)^2 - 7$ By Starting With The Graph Of $y = X^2$ And Using Transformations (shifting, Stretching/compressing, And/or Reflecting).

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Introduction

Graphing functions can be a complex task, but with the right approach, it can be made easier. One of the most effective ways to graph functions is by using transformations. Transformations involve shifting, stretching, compressing, and/or reflecting a parent function to obtain the desired graph. In this article, we will explore how to graph the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7 by starting with the graph of y=x2y = x^2 and using transformations.

Understanding the Parent Function

The parent function in this case is y=x2y = x^2. This is a quadratic function that opens upwards, with its vertex at the origin (0, 0). The graph of this function is a parabola that is symmetric about the y-axis.

Identifying the Transformations

To graph the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7, we need to identify the transformations that need to be applied to the parent function y=x2y = x^2. The given function can be rewritten as f(x)=(x+2)2−7=(x−(−2))2−7f(x) = (x+2)^2 - 7 = (x-(-2))^2 - 7. This means that the parent function has been shifted 2 units to the left and 7 units downwards.

Shifting the Parent Function

The first transformation that needs to be applied is the horizontal shift. Since the parent function has been shifted 2 units to the left, we need to replace xx with x+2x+2 in the parent function. This gives us f(x)=(x+2)2−7f(x) = (x+2)^2 - 7. The graph of this function will be the same as the graph of the parent function, but shifted 2 units to the left.

Reflecting the Parent Function

The second transformation that needs to be applied is the reflection. Since the parent function has been shifted 2 units to the left, we need to reflect the graph of the parent function about the y-axis. This will give us the graph of the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.

Stretching/Compressing the Parent Function

The third transformation that needs to be applied is the stretching/compressing. Since the parent function has been shifted 7 units downwards, we need to compress the graph of the parent function vertically by a factor of 1/7. This will give us the graph of the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.

Graphing the Function

Now that we have identified the transformations that need to be applied, we can graph the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7. The graph of this function will be a parabola that is symmetric about the y-axis, shifted 2 units to the left, and 7 units downwards.

Conclusion

Graphing functions through transformations is a powerful tool that can be used to graph a wide range of functions. By starting with a parent function and applying transformations such as shifting, stretching/compressing, and/or reflecting, we can obtain the desired graph. In this article, we have explored how to graph the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7 by starting with the graph of y=x2y = x^2 and using transformations.

Step-by-Step Guide to Graphing the Function

Here is a step-by-step guide to graphing the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7:

  1. Start with the parent function: Begin by graphing the parent function y=x2y = x^2.
  2. Apply the horizontal shift: Replace xx with x+2x+2 in the parent function to obtain f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.
  3. Apply the reflection: Reflect the graph of the parent function about the y-axis to obtain the graph of the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.
  4. Apply the stretching/compressing: Compress the graph of the parent function vertically by a factor of 1/7 to obtain the graph of the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.
  5. Graph the function: Graph the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7.

Example Problems

Here are some example problems that you can try to practice graphing functions through transformations:

  • Graph the function f(x)=(x−3)2+2f(x) = (x-3)^2 + 2 by starting with the graph of y=x2y = x^2 and using transformations.
  • Graph the function f(x)=(x+1)2−4f(x) = (x+1)^2 - 4 by starting with the graph of y=x2y = x^2 and using transformations.
  • Graph the function f(x)=(x−2)2+1f(x) = (x-2)^2 + 1 by starting with the graph of y=x2y = x^2 and using transformations.

Practice Problems

Here are some practice problems that you can try to practice graphing functions through transformations:

  • Graph the function f(x)=(x+4)2−3f(x) = (x+4)^2 - 3 by starting with the graph of y=x2y = x^2 and using transformations.
  • Graph the function f(x)=(x−1)2+5f(x) = (x-1)^2 + 5 by starting with the graph of y=x2y = x^2 and using transformations.
  • Graph the function f(x)=(x+3)2−2f(x) = (x+3)^2 - 2 by starting with the graph of y=x2y = x^2 and using transformations.

Conclusion

Introduction

Graphing functions through transformations is a powerful tool that can be used to graph a wide range of functions. By starting with a parent function and applying transformations such as shifting, stretching/compressing, and/or reflecting, we can obtain the desired graph. In this article, we will answer some common questions about graphing functions through transformations.

Q: What is the parent function?

A: The parent function is the basic function that we start with when graphing a function through transformations. In the case of the function f(x)=(x+2)2−7f(x) = (x+2)^2 - 7, the parent function is y=x2y = x^2.

Q: What are the different types of transformations?

A: There are three main types of transformations that we can apply to a function:

  • Shifting: This involves moving the graph of the function to the left or right by a certain number of units.
  • Stretching/Compressing: This involves stretching or compressing the graph of the function vertically or horizontally by a certain factor.
  • Reflecting: This involves reflecting the graph of the function about the x-axis or y-axis.

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

A: To apply a horizontal shift to a function, we need to replace xx with x+cx+c in the function, where cc is the number of units that we want to shift the graph to the left or right.

Q: How do I apply a vertical shift to a function?

A: To apply a vertical shift to a function, we need to add or subtract a constant value from the function.

Q: How do I apply a stretching/compressing transformation to a function?

A: To apply a stretching/compressing transformation to a function, we need to multiply the function by a constant factor.

Q: How do I apply a reflecting transformation to a function?

A: To apply a reflecting transformation to a function, we need to replace xx with −x-x or yy with −y-y in the function.

Q: What are some common mistakes to avoid when graphing functions through transformations?

A: Some common mistakes to avoid when graphing functions through transformations include:

  • Not identifying the parent function: Make sure to identify the parent function before applying transformations.
  • Not applying the transformations correctly: Make sure to apply the transformations correctly, including the order in which they are applied.
  • Not checking the graph: Make sure to check the graph to ensure that it is correct.

Q: How can I practice graphing functions through transformations?

A: There are several ways to practice graphing functions through transformations, including:

  • Graphing functions by hand: Practice graphing functions by hand to get a feel for how the transformations work.
  • Using graphing software: Use graphing software such as Desmos or GeoGebra to graph functions and explore the transformations.
  • Solving problems: Practice solving problems that involve graphing functions through transformations.

Q: What are some real-world applications of graphing functions through transformations?

A: Graphing functions through transformations has many real-world applications, including:

  • Physics: Graphing functions through transformations is used to model the motion of objects in physics.
  • Engineering: Graphing functions through transformations is used to design and analyze systems in engineering.
  • Computer Science: Graphing functions through transformations is used to model and analyze algorithms in computer science.

Conclusion

Graphing functions through transformations is a powerful tool that can be used to graph a wide range of functions. By starting with a parent function and applying transformations such as shifting, stretching/compressing, and/or reflecting, we can obtain the desired graph. In this article, we have answered some common questions about graphing functions through transformations.