Question

Function g is a transformation of the parent function f(x)=x^2. The graph of g is translated right 2 units and up 3 units from the graph of f, and then reflected across the y-axis. Write the equation for g in the form y=ax^2+bx+c

Answer 1

Let's understand 4 basic translation rules first:

Let parent function be f(x), so

1. h(x+a) would be f(x) translated a units left

2. h(x-a) would be f(x) translated a units right

3. h(x) + a would be f(x) translated a units up

4. h(x) - a would be f(x) translated a units down

As for reflection, the rules we would need to knw would be:

1. -f(x) would be f(x) replected in x axis

2. f(-x) would be f(x) reflected in y-axis

Now,

given

g is translated 2 units right, that would make f(x):

$\begin{gathered} g(x)=f(x-2)^{} \\ =(x-2)^2 \end{gathered}$

Then,

g is translated 3 units up, so it would be:

$g(x)=(x-2)^2+3$

Last step is to reflect across y-axis.

This means put "-x" in place of x to get:

$\begin{gathered} (x-2)^2+3 \\ =(-x-2)^2+3 \\ =(x-2)^2+3 \end{gathered}$

Same thing.

To write the equation in the form wanted, we multiply:

$\begin{gathered} (x-2)^2+3 \\ =x^2-4x+4+3 \\ =x^2-4x+7 \end{gathered}$

Note:

$(a-b)^2=a^2-2ab+b^2$