Question

What is the equation of the translated function, g(x), if
f(x) = x2?

Answer 1

Let the given function $f(x)$ is

$\quad f(x)=x^{2}+1$

Translation positive $d$ units.

Then, $g(x)=f(x+d)=(x+d)^{2}+1$

$\Rightarrow g(x)=x^{2}+2dx+d^{2}+1$

Translate negative $d$ units.

Then, $g(x)=f(x-d)=(x-d)^{2}+1$

$\Rightarrow g(x)=x^{2}-2dx+d^{2}+1$

Translate negative.

Then, $g(x)=f(-x)=(-x)^{2}+1$

$\Rightarrow g(x)=x^{2}+1$

Translation multiplication.

Then, $g(x)=f(ax)=(ax)^{2}+1$

$\Rightarrow g(x)=a^{2}x^{2}+1$

Remark.

When the question asks for translation, say at $3$, then replace $x$ by $(x+3)$ in the given function and simplify. The required translated function will be found.