Question

The graph of y=(x + 2)^2 – 1 is reflected across the x axis and then translated up 3 units and right 4 units. What is the equation for the transformed graph?

Answer 1

`y=-(x-2)^2\text{ + 4}`

Step-by-step explanation

We have that the graph of y is:

`y=(x+2)^2\text{ - 1}`

It is first reflected about the x axis.

A reflection about the x axis is represented as:

y = -f(x)

which means that we find the negative of the function:

`\begin{gathered} \Rightarrow y=-\lbrack(x+2)^2\text{ - 1\rbrack} \\ y=-(x+2)^2\text{ + 1} \end{gathered}`

Then, it is translated 3 units up (vertical shift) and 4 units right (horizontal shift).

A translation is represented as:

y = f(x - a) + b

where a = horizontal shift; b = vertical shift

So, we have to find:

y = f(x - 4) + 3

That is:

`\begin{gathered} y\text{ = }-\lbrack(x-4)+2\rbrack^2\text{ + 1 + 3} \\ y=-(x-4+2)^2\text{ + 4} \\ y=-(x-2)^2\text{ + 4} \end{gathered}`

Therefore, that is the equation of the transformed graph.