The function f(x)=x^2 is stretched vertically by a factor of 3, translated 2 units to the right, and translated 3 units down. Write the equation of the transformed function.

The function of the graph of f(x)=x² after the graph is stretched vertically by a factor of 3, translated 2 units to the right, and translated 3 units down is f(x) = [3(x-2)²] - 3. This is obtained by using rules of transformation of function.

What are the Rules of Transformation of Function?

Rules of transformation of function are

f(x)+b - function is shifted b units upward

f(x)-b - function is shifted b units downward

f(x+b) - function is shifted b units to the left

f(x-b) - function is shifted b units to the right

-f(x) - function is reflected over x-axis

f(-x) - function is reflected over y-axis
bf(x) - vertical stretch for |b|>1, vertical compression for 0<|b|<1
f(bx) - horizontal compression for |b|>1, horizontal stretch for 0<|b|<1

Find the function required:

Given that the function is f(x)=x²

First the graph is stretched vertically by a factor of 3 units

By the transformation we can rewrite the function in bf(x) form;

that is ⇒f(x) = 3x²

Next the graph is translated 2 units to the right

By the transformation we can rewrite the function in f(x-b) form;

that is ⇒f(x) = 3(x-2)²

Finally the graph is translated 3 units down

By the transformation we can rewrite the function in f(x)-b form;

that is ⇒f(x) = [3(x-2)²] - 3

This is the required function.

Hence the function of the graph of f(x)=x² after the graph is stretched vertically by a factor of 3, translated 2 units to the right, and translated 3 units down is f(x) = [3(x-2)²] - 3.

Learn more about transformation rules here:

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