A Function F ( X ) = X F(x)=\sqrt{x} F ( X ) = X ​ Is Transformed To H ( X ) = − 1 2 X − 3 H(x)=-\frac{1}{2} \sqrt{x-3} H ( X ) = − 2 1 ​ X − 3 ​ . What Transformations Were Used?A. Vertical Stretch By 2, Reflection Over X-axis, Right 3 UnitsB. Vertical Compression By 1 2 \frac{1}{2} 2 1 ​ , Reflection

Introduction

In mathematics, functions can be transformed in various ways to create new functions that exhibit different characteristics. These transformations can be used to model real-world phenomena, solve problems, and gain a deeper understanding of mathematical concepts. In this article, we will explore the transformations of functions, focusing on the given example of $f(x)=\sqrt{x}$ transformed to $h(x)=-\frac{1}{2} \sqrt{x-3}$.

Understanding Function Transformations

Function transformations involve changing the graph of a function in various ways, such as shifting, stretching, compressing, or reflecting. These transformations can be represented algebraically by modifying the function's equation. To identify the transformations used in the given example, we need to analyze the changes made to the original function $f(x)=\sqrt{x}$.

Step 1: Horizontal Shift

The first transformation to consider is the horizontal shift. In the given function $h(x)=-\frac{1}{2} \sqrt{x-3}$, we notice that the argument of the square root function is $x-3$, which means that the graph of the function has been shifted 3 units to the right. This is because the value of $x$ is now $3$ units less than the original value.

Step 2: Reflection

The next transformation to consider is the reflection. In the given function $h(x)=-\frac{1}{2} \sqrt{x-3}$, we notice that the coefficient of the square root function is $-\frac{1}{2}$, which is negative. This indicates that the graph of the function has been reflected over the x-axis.

Step 3: Vertical Compression

The final transformation to consider is the vertical compression. In the given function $h(x)=-\frac{1}{2} \sqrt{x-3}$, we notice that the coefficient of the square root function is $-\frac{1}{2}$, which is less than 1. This indicates that the graph of the function has been compressed vertically by a factor of $\frac{1}{2}$.

Conclusion

In conclusion, the transformations used to transform the function $f(x)=\sqrt{x}$ to $h(x)=-\frac{1}{2} \sqrt{x-3}$ are:

• Horizontal shift 3 units to the right
• Reflection over the x-axis
• Vertical compression by a factor of $\frac{1}{2}$

These transformations can be represented algebraically by modifying the function's equation. By understanding and applying these transformations, we can create new functions that exhibit different characteristics and model real-world phenomena.

Example Problems

To reinforce your understanding of function transformations, try solving the following example problems:

1. What transformations were used to transform the function $f(x)=x^2$ to $h(x)=2(x-1)^2$?
2. What transformations were used to transform the function $f(x)=\sqrt{x}$ to $h(x)=\frac{1}{2} \sqrt{x+2}$?

Answer Key

1. Horizontal shift 1 unit to the right, vertical stretch by a factor of 2
2. Horizontal shift 2 units to the left, vertical compression by a factor of $\frac{1}{2}$

Final Thoughts

Introduction

In our previous article, we explored the transformations of functions, focusing on the given example of $f(x)=\sqrt{x}$ transformed to $h(x)=-\frac{1}{2} \sqrt{x-3}$. In this article, we will continue to delve into the world of function transformations, providing a Q&A guide to help you better understand and apply these concepts.

Q&A: Function Transformations

Q1: What is a horizontal shift in function transformations?

A1: A horizontal shift is a transformation that moves the graph of a function to the left or right. If the graph is shifted to the right, the value of x is increased, and if the graph is shifted to the left, the value of x is decreased.

Q2: How do you represent a horizontal shift algebraically?

A2: A horizontal shift can be represented algebraically by modifying the function's equation. For example, if the graph of a function is shifted 3 units to the right, the new equation would be $h(x) = f(x-3)$.

Q3: What is a vertical stretch in function transformations?

A3: A vertical stretch is a transformation that increases the height of the graph of a function. If the graph is stretched vertically, the value of y is increased.

Q4: How do you represent a vertical stretch algebraically?

A4: A vertical stretch can be represented algebraically by multiplying the function's equation by a constant. For example, if the graph of a function is stretched vertically by a factor of 2, the new equation would be $h(x) = 2f(x)$.

Q5: What is a reflection in function transformations?

A5: A reflection is a transformation that flips the graph of a function over a line. If the graph is reflected over the x-axis, the value of y is negated.

Q6: How do you represent a reflection algebraically?

A6: A reflection can be represented algebraically by multiplying the function's equation by a negative constant. For example, if the graph of a function is reflected over the x-axis, the new equation would be $h(x) = -f(x)$.

Q7: What is a vertical compression in function transformations?

A7: A vertical compression is a transformation that decreases the height of the graph of a function. If the graph is compressed vertically, the value of y is decreased.

Q8: How do you represent a vertical compression algebraically?

A8: A vertical compression can be represented algebraically by multiplying the function's equation by a constant less than 1. For example, if the graph of a function is compressed vertically by a factor of $\frac{1}{2}$, the new equation would be $h(x) = \frac{1}{2}f(x)$.

Q9: How do you determine the type of transformation used to transform a function?

A9: To determine the type of transformation used to transform a function, you need to analyze the changes made to the original function's equation. Look for horizontal shifts, vertical stretches or compressions, and reflections.

Q10: What are some common applications of function transformations?

A10: Function transformations have many applications in mathematics, science, and engineering. Some common applications include modeling population growth, analyzing electrical circuits, and solving optimization problems.

Example Problems

To reinforce your understanding of function transformations, try solving the following example problems:

1. What transformations were used to transform the function $f(x)=x^2$ to $h(x)=2(x-1)^2$?
2. What transformations were used to transform the function $f(x)=\sqrt{x}$ to $h(x)=\frac{1}{2} \sqrt{x+2}$?

Answer Key

1. Horizontal shift 1 unit to the right, vertical stretch by a factor of 2
2. Horizontal shift 2 units to the left, vertical compression by a factor of $\frac{1}{2}$

Final Thoughts

Function transformations are a powerful tool in mathematics, allowing us to create new functions that exhibit different characteristics and model real-world phenomena. By understanding and applying these transformations, we can gain a deeper understanding of mathematical concepts and solve problems more effectively.