How Will The Graph Of $f(x)=\sqrt{x}$ Change If The Function Is Changed To $f(x)=-\frac{1}{3} \sqrt{x}$?A. The Graph Will Stretch Vertically By A Factor Of $\frac{1}{3}$ And Be Reflected About The X-axis.B. The Graph Will

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Introduction

In mathematics, functions are used to describe the relationship between variables. When a function is transformed, its graph undergoes significant changes. In this article, we will explore how the graph of f(x)=xf(x)=\sqrt{x} changes when the function is modified to f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x}. We will delve into the effects of vertical stretching and reflection on the graph.

Original Function: f(x)=xf(x)=\sqrt{x}

The original function is f(x)=xf(x)=\sqrt{x}. This function represents a square root function, where the output is the square root of the input. The graph of this function is a curve that starts at the origin (0,0) and increases as the input value increases.

Key Features of the Original Graph

  • The graph starts at the origin (0,0).
  • The graph increases as the input value increases.
  • The graph is a curve, not a straight line.

Transformed Function: f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x}

The transformed function is f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x}. This function represents a modified square root function, where the output is the negative of one-third times the square root of the input. The graph of this function will undergo significant changes compared to the original graph.

Key Features of the Transformed Graph

  • The graph will be reflected about the x-axis due to the negative sign.
  • The graph will be vertically stretched by a factor of 3 due to the 13\frac{1}{3} coefficient.

Vertical Stretching

When a function is multiplied by a constant, the graph of the function is vertically stretched or compressed by a factor equal to the absolute value of the constant. In this case, the graph of f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x} will be vertically stretched by a factor of 3.

Effect of Vertical Stretching

  • The graph will be stretched vertically by a factor of 3.
  • The graph will be taller and thinner compared to the original graph.

Reflection About the X-Axis

When a function is multiplied by a negative constant, the graph of the function is reflected about the x-axis. In this case, the graph of f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x} will be reflected about the x-axis.

Effect of Reflection

  • The graph will be reflected about the x-axis.
  • The graph will be a mirror image of the original graph.

Combining Vertical Stretching and Reflection

When a function is both vertically stretched and reflected about the x-axis, the graph undergoes both transformations simultaneously. In this case, the graph of f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x} will be both vertically stretched by a factor of 3 and reflected about the x-axis.

Effect of Combining Transformations

  • The graph will be vertically stretched by a factor of 3.
  • The graph will be reflected about the x-axis.
  • The graph will be a vertically stretched and reflected version of the original graph.

Conclusion

In conclusion, the graph of f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x} will undergo significant changes compared to the original graph of f(x)=xf(x)=\sqrt{x}. The graph will be vertically stretched by a factor of 3 and reflected about the x-axis. Understanding these transformations is essential in mathematics, as it allows us to analyze and interpret the behavior of functions and their graphs.

Final Answer

The final answer is: A. The graph will stretch vertically by a factor of 13\frac{1}{3} and be reflected about the x-axis.

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Q1: What is the effect of multiplying a function by a constant on its graph?

A1: When a function is multiplied by a constant, the graph of the function is vertically stretched or compressed by a factor equal to the absolute value of the constant. If the constant is negative, the graph is also reflected about the x-axis.

Q2: How does the graph of f(x)=xf(x)=\sqrt{x} change when the function is modified to f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x}?

A2: The graph of f(x)=−13xf(x)=-\frac{1}{3} \sqrt{x} will be vertically stretched by a factor of 3 and reflected about the x-axis compared to the original graph of f(x)=xf(x)=\sqrt{x}.

Q3: What is the effect of vertical stretching on a graph?

A3: Vertical stretching causes the graph to become taller and thinner. The factor of stretching determines the amount of stretching.

Q4: What is the effect of reflection about the x-axis on a graph?

A4: Reflection about the x-axis causes the graph to be a mirror image of the original graph. The graph is flipped over the x-axis.

Q5: How do you combine vertical stretching and reflection about the x-axis on a graph?

A5: When a function is both vertically stretched and reflected about the x-axis, the graph undergoes both transformations simultaneously. The graph will be a vertically stretched and reflected version of the original graph.

Q6: What is the difference between vertical stretching and horizontal stretching on a graph?

A6: Vertical stretching causes the graph to become taller and thinner, while horizontal stretching causes the graph to become wider and narrower.

Q7: How do you determine the factor of stretching on a graph?

A7: The factor of stretching is determined by the absolute value of the constant that is multiplied by the function.

Q8: What is the effect of a negative constant on a graph?

A8: A negative constant causes the graph to be reflected about the x-axis.

Q9: How do you analyze the behavior of a function and its graph?

A9: To analyze the behavior of a function and its graph, you need to understand the transformations that the function undergoes, such as vertical stretching and reflection about the x-axis.

Q10: Why is it essential to understand function transformations in mathematics?

A10: Understanding function transformations is essential in mathematics because it allows us to analyze and interpret the behavior of functions and their graphs. This knowledge is crucial in various fields, such as physics, engineering, and economics.

Conclusion

In conclusion, understanding function transformations is crucial in mathematics. By analyzing the effects of vertical stretching and reflection about the x-axis, we can gain a deeper understanding of the behavior of functions and their graphs. This knowledge is essential in various fields and can be applied to real-world problems.

Final Answer

The final answer is: Understanding function transformations is essential in mathematics.