How can I find the equation of a transformed function?

Question

For example, if I have a function $f(x) = x^2 + 4x + 1$ and I am transforming it using an equation like $p(x) = 2f(x + 1) - 5$, how would I find the equation of p(x)? I understand the rules of transformation, but do not understand how to apply these to finding transformations of equations. Using Photomath, I get that $p(x) = 2x^2 + 12x + 7$, but I don’t understand why this is.

Answer 1

$p(x) = 2f(x+1) - 5$

Let's say for a moment that $u = x+1.$ This step is not really necessary, but it might help make the details a little more clear.

$p(x) = 2f(u) - 5$

Put in what we know for $f(u)$

$p(x) = 2(u^2 + 4u+1) - 5$

Reverse the substitution.

$p(x) = 2((x+1)^2 + 4(x+1) + 1) - 5.$ If you were to have jumped to this place, that would be fine.

Multiply out what we have and simplify.

$p(x) = 2(x^2 + 2x +1 + 4x+4 + 1) - 5\\ 2(x^2 + 6x +6) - 5\\ 2x^2 + 12x +12 - 5\\ 2x^2 + 12x +7$