Analyze The Function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ And Sketch The Graph.Hint: When $x$ Is Approaching Zero, $\frac{\sqrt{2}(x-2)^2}{2} \sim 2^2 \sqrt{x} / 2$. Otherwise, The Parabola $\sqrt{2}(x-2)^2 / 2$

Introduction

In this article, we will delve into the analysis of the given function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ and sketch its graph. The function appears to be a quadratic function, but with a twist. We will explore its behavior as $x$ approaches zero and its general behavior otherwise.

Understanding the Function

The given function is $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$. At first glance, it seems like a quadratic function, but with a square root term. Let's break it down further.

• Quadratic Term: The term $(x-2)^2$ is a quadratic term, which is a polynomial of degree 2. It represents a parabola that opens upwards or downwards.
• Square Root Term: The term $\sqrt{2}$ is a square root term, which is a non-polynomial term. It represents a non-polynomial function that grows or decays exponentially.
• Constant Term: The term $\frac{1}{2}$ is a constant term, which is a polynomial of degree 0. It represents a horizontal line.

Behavior as $x$ Approaches Zero

As $x$ approaches zero, the function $f(x)$ behaves differently. We can use the hint provided to analyze its behavior.

• Approximation: When $x$ is approaching zero, we can approximate the function $f(x)$ as $\frac{\sqrt{2}(x-2)^2}{2} \sim 2^2 \sqrt{x} / 2$.
• Behavior: As $x$ approaches zero, the function $f(x)$ approaches zero as well. This is because the quadratic term $(x-2)^2$ approaches zero, and the square root term $\sqrt{2}$ remains constant.

General Behavior

As $x$ moves away from zero, the function $f(x)$ behaves differently. We can analyze its behavior by considering the general form of the function.

• Parabola: The function $f(x)$ is a parabola that opens upwards. This is because the quadratic term $(x-2)^2$ is always positive, and the square root term $\sqrt{2}$ is always positive.
• Vertex: The vertex of the parabola is at $x=2$. This is because the quadratic term $(x-2)^2$ is zero at $x=2$, and the square root term $\sqrt{2}$ remains constant.

Sketching the Graph

To sketch the graph of the function $f(x)$, we can use the following steps:

1. Plot the Vertex: Plot the vertex of the parabola at $x=2$.
2. Plot the Asymptote: Plot the asymptote of the parabola, which is the horizontal line $y=0$.
3. Plot the Parabola: Plot the parabola that opens upwards, using the general form of the function.

Conclusion

In conclusion, the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is a quadratic function with a twist. As $x$ approaches zero, the function behaves differently, and as $x$ moves away from zero, the function behaves like a parabola that opens upwards. We can sketch the graph of the function by plotting the vertex, asymptote, and parabola.

References

• [1] "Calculus" by Michael Spivak
• [2] "Algebra" by Michael Artin

Further Reading

For further reading on the topic of quadratic functions and their graphs, we recommend the following resources:

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

Glossary

• Quadratic Function: A polynomial function of degree 2.
• Square Root Term: A non-polynomial term that grows or decays exponentially.
• Constant Term: A polynomial of degree 0.
• Vertex: The point on the graph of a parabola where the function changes from increasing to decreasing or vice versa.
• Asymptote: A horizontal or vertical line that the graph of a function approaches but never touches.
  Q&A: Analyzing the Function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ and Sketching the Graph =====================================================================================

Introduction

In our previous article, we analyzed the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ and sketched its graph. In this article, we will answer some frequently asked questions about the function and its graph.

Q: What is the domain of the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$?

A: The domain of the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is all real numbers, except for $x=2$. This is because the function is undefined at $x=2$, where the denominator is zero.

Q: What is the range of the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$?

A: The range of the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is all non-negative real numbers. This is because the function is always non-negative, and it approaches zero as $x$ approaches infinity.

Q: What is the vertex of the parabola represented by the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$?

A: The vertex of the parabola represented by the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is at $x=2$. This is because the quadratic term $(x-2)^2$ is zero at $x=2$, and the square root term $\sqrt{2}$ remains constant.

Q: What is the asymptote of the parabola represented by the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$?

A: The asymptote of the parabola represented by the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is the horizontal line $y=0$. This is because the function approaches zero as $x$ approaches infinity.

Q: How does the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ behave as $x$ approaches zero?

A: As $x$ approaches zero, the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ approaches zero as well. This is because the quadratic term $(x-2)^2$ approaches zero, and the square root term $\sqrt{2}$ remains constant.

Q: How does the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ behave as $x$ approaches infinity?

A: As $x$ approaches infinity, the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ approaches infinity as well. This is because the quadratic term $(x-2)^2$ approaches infinity, and the square root term $\sqrt{2}$ remains constant.

Conclusion

In conclusion, the function $f(x)=\frac{\sqrt{2}(x-2)^2}{2}$ is a quadratic function with a twist. We have answered some frequently asked questions about the function and its graph, and we hope that this article has been helpful in understanding the function and its behavior.

References

• [1] "Calculus" by Michael Spivak
• [2] "Algebra" by Michael Artin

Further Reading

For further reading on the topic of quadratic functions and their graphs, we recommend the following resources:

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

Glossary

• Quadratic Function: A polynomial function of degree 2.
• Square Root Term: A non-polynomial term that grows or decays exponentially.
• Constant Term: A polynomial of degree 0.
• Vertex: The point on the graph of a parabola where the function changes from increasing to decreasing or vice versa.
• Asymptote: A horizontal or vertical line that the graph of a function approaches but never touches.