Write a formula for g(x), a function whose graph looks like the followin 2- -4 -3 -2 -1 2 3 4. -2- -4 9- NOTE: Your answer should have f (x), appropriately transformed, in it!
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.
![**Understanding Transformations of Functions**
Consider a function \( f(x) \) whose graph is given below:
(Graph of \( f(x) \))
The graph provided shows the function \(f(x)\) which forms a V-shape. The V-shape is symmetric about the y-axis with the vertex of the V at the point (0, 3). The left arm of the V descends linearly from (0, 3) to (-3, 0), and the right arm descends linearly from (0, 3) to (3, 0).
### Original Function \( f(x) \)
- Vertex: (0,3)
- When \(x = -3\), \(f(x) = 0\)
- When \(x = 3\), \(f(x) = 0\)
The graph appears to be a transformed absolute value function with adjustments to height.
### Task: Write a formula for \( g(x) \)
Now, let's consider a new function \( g(x) \) whose graph is provided below:
(Graph of \( g(x) \))
The given graph of \( g(x) \) indicates a similar V-shape, but the vertex has been shifted up to the point (0, 6), and the arms descend linearly from the vertex to \( (-3, 0) \) and \( (3, 0) \) respectively.
### Transformed Function \( g(x) \)
Based on visual analysis and comparison:
- The function \( g(x) \) appears to be a vertically stretched version of \( f(x) \) by a factor of 2.
- Finally, add 3 to the vertex to shift it from 3 to 6.
Thus,
\[ g(x) = 2f(x) - 3 \]
### Note:
Ensure your final answer contains \( f(x) \) appropriately transformed for accurate representation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3d88c57-73ad-4bc1-a9bf-667243851eff%2F8afaa8f7-fa58-4428-a50a-93fbe2da848c%2Fwg4isas_processed.jpeg&w=3840&q=75)
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