shifted right 3 units and shifted down 4 units. O f(z) = Va + 3 - 4 O f(z) = VI -3- 4 O f(z) = Vz + 3 + 4 O f(z) = VI – 3 + 4 The graph of f(x) stretched vertically by a factor of 5. O f(z) = 5/T O f(2) = VE %3D O f(z) = V5x O f(x) =
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.
![### Transformations of Functions
This content is intended for educational purposes to help students understand the transformations of functions, specifically the square root function \( f(x) = \sqrt{x} \).
#### Horizontal and Vertical Shifting
**Original Function:**
\[ f(x) = \sqrt{x} \]
**Transformed Functions:**
1. **Shift Right 3 Units and Down 4 Units:**
- To shift the graph of \( f(x) = \sqrt{x} \) right by 3 units, replace \( x \) with \( x-3 \).
- To shift it down by 4 units, subtract 4 from the entire function.
- This results in the transformed function:
\[ f(x) = \sqrt{x - 3} - 4 \]
**Multiple Choice Answers (one correct):**
- \( \circ f(x) = \sqrt{x + 3} - 4 \)
- \( \circ f(x) = \sqrt{x - 3} - 4 \) ⬅️ Correct Answer
- \( \circ f(x) = \sqrt{x + 3} + 4 \)
- \( \circ f(x) = \sqrt{x - 3} + 4 \)
#### Vertical Stretching or Shrinking
**Original Function:**
\[ f(x) = \sqrt{x} \]
**Transformed Functions:**
2. **Stretch Vertically by a Factor of 5:**
- To stretch the graph of \( f(x) = \sqrt{x} \) vertically by a factor of 5, multiply the entire function by 5.
- This results in the transformed function:
\[ f(x) = 5\sqrt{x} \]
**Multiple Choice Answers (one correct):**
- \( \circ f(x) = 5\sqrt{x} \) ⬅️ Correct Answer
- \( \circ f(x) = \frac{1}{5}\sqrt{x} \)
- \( \circ f(x) = \sqrt{5x} \)
- \( \circ f(x) = \sqrt{\frac{1}{5}}x \)
#### Summary
In this lesson, you learned how to apply horizontal and vertical shifts to the square root function, as well as how to perform vertical stretching. Transformations such as these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0cf52f6a-9168-49cc-a8d7-0312a37994ea%2F75ee1bd1-4136-409b-a69e-9caf2a5d64f7%2F9kvntjd_processed.jpeg&w=3840&q=75)
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