MHB Multiple Transformations of Functions

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To transform the function f(x) = x^3 into the new function, a horizontal shift of 3 units to the right and a vertical shift of 1 unit upward is required. The transformation can be expressed as f(x) = a(x-3)^3 + 1, where 'a' represents the stretch factor. Given the point (4, 1.5), the value of 'a' can be determined by substituting this point into the equation. This results in an equation that allows for the calculation of 'a', completing the transformation process. Understanding these transformations is crucial for accurately modifying the function.
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I have to transform the first function which is f(x)=x^3 to the second function. First, I have to find each shift then combine those to make a new function equation. I've used desmos and I know that there is a horizontal shift 3 units to the right. There is a vertical shift up but I don't know how many units. And I believe there is a stretch. There are only 3 transformations. PLEASE HELP!
 

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note the function center point $(0,0)$ is translated to $(3,1)$, a horizontal shift right 3 units and a vertical shift up 1 unit.

taking into account the horizontal & vertical shifts, we have ...

$f(x) = a(x-3)^3 + 1$

... where $a$ is the constant causing the stretch

using the point $(4,1.5)$, can you determine the value of $a$ ?
 

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Thread 'Erroneously finding discrepancy in transpose rule'

Thread 'Erroneously finding discrepancy in transpose rule'

Obviously, there is something elementary I am missing here. To form the transpose of a matrix, one exchanges rows and columns, so the transpose of a scalar, considered as (or isomorphic to) a one-entry matrix, should stay the same, including if the scalar is a complex number. On the other hand, in the isomorphism between the complex plane and the real plane, a complex number a+bi corresponds to a matrix in the real plane; taking the transpose we get which then corresponds to a-bi...
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