# Multiple Transformations of Functions

• MHB
In summary, the conversation discusses transforming the function f(x)=x^3 to a new function by finding the shifts and combining them. The resulting function is f(x) = a(x-3)^3 + 1, where a is the constant causing the stretch. The point (4,1.5) can be used to determine the value of a.

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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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• cubic_transformation.jpg
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## What is a transformation of a function?

A transformation of a function is a change made to the original function that affects its graph. This can include shifting, stretching, or reflecting the graph.

## How do you identify multiple transformations of a function?

To identify multiple transformations of a function, you must first identify the original function and then look for any changes made to it. These changes can be in the form of added or subtracted values, coefficients, or variables.

## What is the order of operations for multiple transformations of functions?

The order of operations for multiple transformations of functions is the same as for mathematical expressions. The transformations are applied in the following order: vertical shifts, horizontal shifts, reflections, and stretches/compressions.

## How do you graph a function with multiple transformations?

To graph a function with multiple transformations, start by graphing the original function. Then, apply each transformation in the order of operations. This will result in a new graph that represents the function with all of its transformations.

## What is the difference between a horizontal and vertical transformation of a function?

A horizontal transformation of a function affects the inputs (x-values) and shifts the graph left or right. A vertical transformation affects the outputs (y-values) and shifts the graph up or down.