So I have the following question: A function $y = f(x) = 2x^2 + x + 1$ undergoes following transformation. A horizontal translation $2$ units to the right, followed by horizontal dilation by factor $\frac{1}{3}$ and then vertical dilation by $4$ units. Find the equation of the transformed curve in the form $y = ax^2 + bx + c$ and state the values of $a,b$ and $c$.
So horizontal translation is easy: I get $2(x-2)^{2} + (x-2) + 1$
Then I am confused about horizontal dilation whether it is $2(3x-2)^{2} + (3x-2) + 1$ or $2(3(x-2))^{2} + (3(x-2)) + 1$.
And for vertical dilation I am confused whether to multiply whole thing by $4$ or $\frac{1}{4}$.
I am confused... please help.
The first expression for the horizontal enlargement is correct because every $x$ has been replaced by $3x$
The second expression you are asking about is actually the same two transformations but in reverse order, i.e. enlargement followed by translation, because in the expression for $f(3x)$ you have replaced every $x$ by $(x-2)$.
For the vertical enlargement by factor $4$ you just multiply the whole expression by $4$.
I hope this helps.