The graph to the right is obtained from the graph of g(x) = Vx by applying several transformations. Describe the transformations and give the equation of the graph. What horizontal translation, applied to the graph of g(x) = Vx before any other transformations, would be needed to obtain the given graph? O A. Translate g(x) | unit(s) to the right. (Type a whole number.) O B. Translate g(x) unit(s) to the left. (Type a whole number.) OC. No horizontal translation is needed. What type of stretching or shrinking, applied to the graph of g(x) = /x after any horizontal translations but before any other transformations, would be needed to obtain the given graph? O A. Vertically shrink g(x) by a factor of O B. Horizontally shrink g(x) by a factor of O C. Horizontally stretch g(x) by a factor of O D. Vertically stretch g(x) by a factor of O E. No stretching or shrinking is needed.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![### Transformation of the Function \( g(x) = \sqrt{x} \)
#### Reflections
**Question:**
What reflection, applied to the graph of \( g(x) = \sqrt{x} \) after any horizontal translations and stretching or shrinking but before any other transformations, would be needed to obtain the given graph?
- **A.** Reflect \( g(x) \) across the x-axis.
- **B.** Reflect \( g(x) \) across the y-axis.
- **C.** No reflection is needed.
#### Vertical Translations
**Question:**
What vertical translation, applied to the graph of \( g(x) = \sqrt{x} \) after all other transformations, would be needed to obtain the given graph?
- **A.** Translate \( g(x) \) \[ \_\_ \] unit(s) up. (Type a whole number.)
- **B.** Translate \( g(x) \) \[ \_\_ \] unit(s) down. (Type a whole number.)
- **C.** No vertical translation is needed.
#### Final Equation
**Question:**
Give the equation of the graph.
\[ \_\_\_\_\_\_\_\_ \] (Type an equation.)
### Instructions
- For the **Reflections** section, choose the appropriate reflection needed for the transformation.
- For the **Vertical Translations** section, determine if any vertical translation is required and specify the number of units up or down.
- Finally, in the **Final Equation** section, type the equation that represents the graph after all transformations have been applied.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffafbea1e-da97-44d9-88cf-94bc0d1ddd25%2F90c81a27-7993-48cc-9ce9-81c2c70dfd0c%2F67ed0nc_processed.png&w=3840&q=75)
![**Analyzing Graph Transformations**
The graph in question is transformed from the parent function \( g(x) = \sqrt{x} \) by applying several modifications. Your task is to describe these transformations accurately and derive the equation of the modified graph.
1. **Graph Analysis and Diagram Explanation:**
- The provided diagram depicts a grid with the x-axis and y-axis marked.
- On the grid, a curve resembling the square root function is plotted.
- The curve appears to start slightly to the right of the y-axis (around \( x = 1 \)), indicating a horizontal shift.
- The curve also seems to rise less steeply than the standard \( \sqrt{x} \) function, suggesting a vertical shrink or stretch has been applied.
2. **Determining Horizontal Translation:**
- Question Prompt: What horizontal translation, applied to the graph of \( g(x) = \sqrt{x} \) before any other transformations, would be needed to obtain the given graph?
- A. Translate \( g(x) \) _______ unit(s) to the right. (Type a whole number.)
- B. Translate \( g(x) \) _______ unit(s) to the left. (Type a whole number.)
- C. No horizontal translation is needed.
3. **Determining Stretching or Shrinking:**
- Question Prompt: What type of stretching or shrinking, applied to the graph of \( g(x) = \sqrt{x} \) after any horizontal translations but before any other transformations, would be needed to obtain the given graph?
- A. Vertically shrink \( g(x) \) by a factor of _______.
- B. Horizontally shrink \( g(x) \) by a factor of _______.
- C. Horizontally stretch \( g(x) \) by a factor of _______.
- D. Vertically stretch \( g(x) \) by a factor of _______.
- E. No stretching or shrinking is needed.
**Conclusion:**
To determine the correct transformations, observe the starting point and the steepness of the curve. First, address the horizontal translation required by comparing the new starting point to the original graph. Then, consider vertical or horizontal scaling necessary to match the shape of the given curve.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffafbea1e-da97-44d9-88cf-94bc0d1ddd25%2F90c81a27-7993-48cc-9ce9-81c2c70dfd0c%2Fj8kjogp_processed.png&w=3840&q=75)
![](/static/compass_v2/shared-icons/check-mark.png)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
![Algebra and Trigonometry (6th Edition)](https://www.bartleby.com/isbn_cover_images/9780134463216/9780134463216_smallCoverImage.gif)
![Contemporary Abstract Algebra](https://www.bartleby.com/isbn_cover_images/9781305657960/9781305657960_smallCoverImage.gif)
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
![Algebra and Trigonometry (6th Edition)](https://www.bartleby.com/isbn_cover_images/9780134463216/9780134463216_smallCoverImage.gif)
![Contemporary Abstract Algebra](https://www.bartleby.com/isbn_cover_images/9781305657960/9781305657960_smallCoverImage.gif)
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
![Algebra And Trigonometry (11th Edition)](https://www.bartleby.com/isbn_cover_images/9780135163078/9780135163078_smallCoverImage.gif)
![Introduction to Linear Algebra, Fifth Edition](https://www.bartleby.com/isbn_cover_images/9780980232776/9780980232776_smallCoverImage.gif)
![College Algebra (Collegiate Math)](https://www.bartleby.com/isbn_cover_images/9780077836344/9780077836344_smallCoverImage.gif)