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.
![### Graphing Rational Functions
#### Problem:
**Task:** Write an equation for the function graphed below. The y-intercept is at (0, 3).
#### Graph Explanation:
The graph provided depicts a rational function characterized by vertical asymptotes, a horizontal asymptote, and a distinctive curve shape. Here's a detailed breakdown:
1. **Vertical Asymptotes:**
- The graph has vertical asymptotes (dotted red lines) at \( x = -2 \) and \( x = 2 \).
2. **Y-Intercept:**
- The y-intercept is given at the coordinate (0, 3).
3. **Curve Behavior:**
- The graph approaches negative infinity as \( x \) approaches -2 from the left and positive infinity as \( x \) approaches -2 from the right.
- Similarly, the graph approaches negative infinity as \( x \) approaches 2 from the left and positive infinity as \( x \) approaches 2 from the right.
- There is a horizontal asymptote around \( y = 0 \) (though it's not explicitly marked).
#### Objective:
To write an equation \( y = \) representing the function based on the observations of the provided graph.
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### Interactive Exercise:
Using the characteristics discussed:
- Identify the roots and asymptotes.
- Formulate the equation for the function.
**Equation Form:**
\[ y = \frac{a}{(x + 2)(x - 2)} \]
Given the y-intercept (0, 3):
\[ 3 = \frac{a}{(0 + 2)(0 - 2)} \Rightarrow 3 = \frac{a}{-4} \Rightarrow a = -12 \]
Therefore:
\[ y = \frac{-12}{(x + 2)(x - 2)} \]
To interact with the graph, input your equation below:
\[ y = \]
Find additional exercises and tutorials on rational functions [here](#).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88d18a19-21f4-41fb-afe0-306734676977%2Ff633be8d-90e0-4ce3-9fea-9e320d1fa398%2Fki1h42y.png&w=3840&q=75)
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