Which Graph Represents The Function F ( X ) = ( X − 5 ) 2 + 3 F(x) = (x-5)^2 + 3 F ( X ) = ( X − 5 ) 2 + 3 ?

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Introduction

In mathematics, graphing functions is a crucial concept that helps us visualize and understand the behavior of various mathematical equations. When it comes to quadratic functions, graphing them can be a bit tricky, but with the right approach, it can be a fun and rewarding experience. In this article, we will explore the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 and determine which graph represents this function.

Understanding the Function

Before we dive into graphing the function, let's take a closer look at its equation. The function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 is a quadratic function in the form of f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where aa, hh, and kk are constants. In this case, a=1a = 1, h=5h = 5, and k=3k = 3. This means that the function has a vertex at the point (5,3)(5, 3) and opens upwards.

Graphing the Function

To graph the function, we can start by plotting the vertex at the point (5,3)(5, 3). Since the function opens upwards, the graph will be a parabola that opens upwards. We can then use the fact that the function is in the form of f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k to determine the direction and shape of the graph.

Key Features of the Graph

The graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 has several key features that we need to consider when determining which graph represents this function. These features include:

  • Vertex: The vertex of the graph is located at the point (5,3)(5, 3).
  • Direction: The graph opens upwards, indicating that the function is increasing as xx increases.
  • Shape: The graph is a parabola, which is a U-shaped curve.
  • Axis of Symmetry: The axis of symmetry of the graph is the vertical line x=5x = 5.

Analyzing the Graphs

Now that we have a good understanding of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3, let's analyze the graphs provided and determine which one represents this function.

Graph A

Graph A is a parabola that opens upwards, with a vertex at the point (5,3)(5, 3). The graph has a minimum point at the vertex and increases as xx increases. This graph matches the key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3.

Graph B

Graph B is a parabola that opens downwards, with a vertex at the point (5,3)(5, 3). The graph has a maximum point at the vertex and decreases as xx increases. This graph does not match the key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3.

Graph C

Graph C is a parabola that opens upwards, with a vertex at the point (5,3)(5, 3). However, the graph has a maximum point at the vertex and decreases as xx increases. This graph does not match the key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3.

Graph D

Graph D is a parabola that opens downwards, with a vertex at the point (5,3)(5, 3). The graph has a minimum point at the vertex and increases as xx increases. This graph does not match the key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3.

Conclusion

Based on the analysis of the graphs provided, we can conclude that Graph A represents the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3. This graph matches the key features of the graph of the function, including the vertex, direction, shape, and axis of symmetry.

Final Thoughts

Graphing functions is an essential concept in mathematics that helps us visualize and understand the behavior of various mathematical equations. By analyzing the key features of a graph, we can determine which graph represents a given function. In this article, we explored the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 and determined which graph represents this function. We hope that this article has provided you with a better understanding of graphing functions and how to analyze graphs to determine which one represents a given function.

References

  • [1] "Graphing Quadratic Functions" by Math Open Reference
  • [2] "Graphing Functions" by Khan Academy
  • [3] "Quadratic Functions" by Purplemath

Additional Resources

  • [1] "Graphing Functions" by Mathway
  • [2] "Quadratic Functions" by IXL
  • [3] "Graphing Quadratic Functions" by Algebra.com

Introduction

In our previous article, we explored the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 and determined which graph represents this function. In this article, we will answer some frequently asked questions (FAQs) about graphing this function.

Q&A

Q: What is the vertex of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: The vertex of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 is located at the point (5,3)(5, 3).

Q: What is the direction of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: The graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 opens upwards, indicating that the function is increasing as xx increases.

Q: What is the shape of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: The graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 is a parabola, which is a U-shaped curve.

Q: What is the axis of symmetry of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: The axis of symmetry of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 is the vertical line x=5x = 5.

Q: How do I graph the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: To graph the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3, start by plotting the vertex at the point (5,3)(5, 3). Then, use the fact that the function is in the form of f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k to determine the direction and shape of the graph.

Q: What are some key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: Some key features of the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 include:

  • Vertex: The vertex of the graph is located at the point (5,3)(5, 3).
  • Direction: The graph opens upwards, indicating that the function is increasing as xx increases.
  • Shape: The graph is a parabola, which is a U-shaped curve.
  • Axis of Symmetry: The axis of symmetry of the graph is the vertical line x=5x = 5.

Q: How do I determine which graph represents the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3?

A: To determine which graph represents the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3, analyze the key features of the graph, including the vertex, direction, shape, and axis of symmetry.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about graphing the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3. We hope that this article has provided you with a better understanding of graphing functions and how to analyze graphs to determine which one represents a given function.

Final Thoughts

Graphing functions is an essential concept in mathematics that helps us visualize and understand the behavior of various mathematical equations. By analyzing the key features of a graph, we can determine which graph represents a given function. In this article, we explored the graph of the function f(x)=(x−5)2+3f(x) = (x-5)^2 + 3 and answered some frequently asked questions (FAQs) about graphing this function.

References

  • [1] "Graphing Quadratic Functions" by Math Open Reference
  • [2] "Graphing Functions" by Khan Academy
  • [3] "Quadratic Functions" by Purplemath

Additional Resources

  • [1] "Graphing Functions" by Mathway
  • [2] "Quadratic Functions" by IXL
  • [3] "Graphing Quadratic Functions" by Algebra.com