Question

Given: f(x) = Describe the transformation of the green and blue graph.

Answer 1

We have a function f(x) that is represented with the black curve.

We have to find the transformation to make this function become the green curve and the blue curve (this will be different transformations for each line).

We can compare the black curve to the green curve: they have the same shape and orientation. We can see that from the initial point of the curve, the point two units up and two units right from the vertex also belongs to the curve.

Then, as the shape and orientation is the same, the only transformation is a translation.

The value of the translation for each coordinate can be find as the difference between corresponding points of both curves (for example, the vertices of each curve):

We can see that the translation is 3 units to the right and 4 units up. This can be written as a rule like:

`T_(<3,4>)\colon(x,y)\longrightarrow(x+3,y+4)`

Then, the transformation for f(x) to become the green line is a translation 3 units to the right and 4 units to the right: T<3,4>.

Now we can compare the black line and the blue line. In this case, the lines have the same shape, but the vertex is translated and the orientation is different: the line is reflected over the x-axis.

We then can think of a translation 2 units to the right and then a reflection over the x-axis:

`\begin{gathered} T_(<-2,0>)\colon(x,y)\longrightarrow(x-2,y) \\ R_{\text{x-axis}}\colon(x-2,y)\longrightarrow(x-2,-y) \end{gathered}`

Then, we can conclude that to transform the black curve into the blue curve, we apply a translation T<-2,0> and a reflection over the x-axis.

Answer:

Green line --> We apply a translation T<3,4> or 3 units right and 4 units up.

Blue line --> We apply a translation T<-2,0>, or 2 units left, and a reflection over the x-axis.