Which Graph Shows The Axis Of Symmetry For The Function $f(x) = (x - 2)^2 + 1$?

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Introduction

When analyzing functions, particularly quadratic functions, it's essential to understand the concept of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts. In this article, we will explore how to identify the axis of symmetry for the function $f(x) = (x - 2)^2 + 1$.

What is the Axis of Symmetry?

The axis of symmetry is a fundamental concept in mathematics, particularly in algebra and geometry. It's a line that passes through the vertex of a parabola, dividing it into two symmetrical parts. The axis of symmetry is denoted by the equation x = a, where 'a' is the x-coordinate of the vertex.

Identifying the Axis of Symmetry

To identify the axis of symmetry for the function $f(x) = (x - 2)^2 + 1$, we need to follow these steps:

  1. Identify the vertex form: The given function is already in vertex form, which is $f(x) = (x - h)^2 + k$, where (h, k) is the vertex of the parabola.
  2. Extract the x-coordinate of the vertex: From the given function, we can see that the x-coordinate of the vertex is h = 2.
  3. Write the equation of the axis of symmetry: The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex.

Graphing the Axis of Symmetry

To graph the axis of symmetry, we need to plot the line x = 2 on the coordinate plane. This line will pass through the vertex of the parabola, dividing it into two symmetrical parts.

Conclusion

In conclusion, the axis of symmetry for the function $f(x) = (x - 2)^2 + 1$ is the line x = 2. This line passes through the vertex of the parabola, dividing it into two symmetrical parts. Understanding the axis of symmetry is essential in analyzing functions, particularly quadratic functions.

Example Problems

Problem 1

Find the axis of symmetry for the function $f(x) = (x + 3)^2 - 4$.

Solution

To find the axis of symmetry, we need to extract the x-coordinate of the vertex. From the given function, we can see that the x-coordinate of the vertex is h = -3. Therefore, the equation of the axis of symmetry is x = -3.

Problem 2

Find the axis of symmetry for the function $f(x) = (x - 5)^2 + 2$.

Solution

To find the axis of symmetry, we need to extract the x-coordinate of the vertex. From the given function, we can see that the x-coordinate of the vertex is h = 5. Therefore, the equation of the axis of symmetry is x = 5.

Tips and Tricks

  • To find the axis of symmetry, we need to extract the x-coordinate of the vertex from the given function.
  • The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex.
  • The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts.

Real-World Applications

Understanding the axis of symmetry has numerous real-world applications, including:

  • Physics: The axis of symmetry is used to describe the motion of objects under the influence of gravity.
  • Engineering: The axis of symmetry is used to design and analyze structures, such as bridges and buildings.
  • Computer Science: The axis of symmetry is used in algorithms and data structures to optimize performance and efficiency.

Conclusion

In conclusion, the axis of symmetry is a fundamental concept in mathematics, particularly in algebra and geometry. Understanding the axis of symmetry is essential in analyzing functions, particularly quadratic functions. By following the steps outlined in this article, we can identify the axis of symmetry for any given function.

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Frequently Asked Questions

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts.

Q: How do I find the axis of symmetry for a given function?

A: To find the axis of symmetry, you need to extract the x-coordinate of the vertex from the given function. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is $f(x) = (x - h)^2 + k$, where (h, k) is the vertex of the parabola.

Q: How do I identify the vertex of a parabola?

A: To identify the vertex of a parabola, you need to look for the minimum or maximum point of the parabola. The vertex is the point where the parabola changes direction.

Q: What is the significance of the axis of symmetry in real-world applications?

A: The axis of symmetry has numerous real-world applications, including physics, engineering, and computer science. It is used to describe the motion of objects under the influence of gravity, design and analyze structures, and optimize algorithms and data structures.

Q: Can the axis of symmetry be used to solve problems in other areas of mathematics?

A: Yes, the axis of symmetry can be used to solve problems in other areas of mathematics, such as calculus and differential equations.

Q: How do I graph the axis of symmetry on a coordinate plane?

A: To graph the axis of symmetry, you need to plot the line x = h on the coordinate plane, where h is the x-coordinate of the vertex.

Q: What is the relationship between the axis of symmetry and the vertex of a parabola?

A: The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two symmetrical parts.

Q: Can the axis of symmetry be used to find the x-intercepts of a parabola?

A: Yes, the axis of symmetry can be used to find the x-intercepts of a parabola. The x-intercepts are the points where the parabola intersects the x-axis.

Q: How do I use the axis of symmetry to solve problems in physics and engineering?

A: To use the axis of symmetry to solve problems in physics and engineering, you need to apply the concept of symmetry to the problem at hand. This may involve using the axis of symmetry to describe the motion of objects under the influence of gravity or to design and analyze structures.

Common Mistakes to Avoid

  • Not extracting the x-coordinate of the vertex: Failing to extract the x-coordinate of the vertex can lead to incorrect identification of the axis of symmetry.
  • Not using the correct equation of the axis of symmetry: Using the wrong equation of the axis of symmetry can lead to incorrect results.
  • Not applying the concept of symmetry correctly: Failing to apply the concept of symmetry correctly can lead to incorrect solutions to problems.

Conclusion

In conclusion, the axis of symmetry is a fundamental concept in mathematics, particularly in algebra and geometry. Understanding the axis of symmetry is essential in analyzing functions, particularly quadratic functions. By following the steps outlined in this article, we can identify the axis of symmetry for any given function and apply it to solve problems in various areas of mathematics.

Additional Resources

  • Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "Calculus" by James Stewart
  • Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
  • Software: Graphing calculators, computer algebra systems

Final Thoughts

Understanding the axis of symmetry is a crucial step in analyzing functions, particularly quadratic functions. By applying the concept of symmetry, we can solve problems in various areas of mathematics and real-world applications. Remember to extract the x-coordinate of the vertex, use the correct equation of the axis of symmetry, and apply the concept of symmetry correctly to achieve accurate results.