Determine the cubic function obtained from the parent function y=x^3 after each sequence of transformations.
a reflection across the x-axis; a translation 1 unit up; and a translation 2 units right

Answer:

-(y+3)=(x-2)^3

Step-by-step explanation:

Transformed function B is f' ( x ) = - ( x - 2 )³ + 1

The transformed function is given by y = - ( x - 2 )³ + 1

How does the transformation of a function happen?

The transformation of a function may involve any change.

Usually, these can be shifted horizontally (by transforming inputs) or vertically (by transforming output), stretched (multiplying outputs or inputs), etc.

If the original function is y = f(x), assuming the horizontal axis is the input axis and the vertical is for outputs, then:

Horizontal shift (also called phase shift):

Left shift by c units:  y=f(x+c) (same output, but c units earlier)

Right shift by c units:  y=f(x-c)(same output, but c units late)

Vertical shift:

Up by d units: y = f(x) + d

Down by d units: y = f(x) - d

Stretching:

Vertical stretch by a factor k: y = k × f(x)

Horizontal stretch by a factor k: y = f(x/k)

Given data ,

Let the parent function be denoted as A

Now , the value of A is

y = x³

Let the transformed function be denoted as B

Now , the value of B is calculated by

Step 1 :

A reflection across the x-axis

So , the function changes to g ( x ) = -x³

Step 2 :

A translation 1 unit up

Up by d units: y = f(x) + d

So , the function changes to h ( x ) = -x³ + 1

Step 3 :

A translation of 2 units right

Right shift by c units:  y=f(x-c)

So , the function changes to f' ( x ) = - ( x - 2 )³ + 1

Therefore , the value of B is f' ( x ) = - ( x - 2 )³ + 1

Hence , the transformed function B is f' ( x ) = - ( x - 2 )³ + 1

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