Thread: Transformation of graph

1. Transformation of graph

A graph with the equation $\displaystyle y=f(x)$ undergoes, in succession, the following transformations:

A: A translation of $\displaystyle 1$ unit in the direction of the positive x-axis.
B: A scaling parallel to the x-axis by a scale factor $\displaystyle \frac{1}{2}$
C: A reflection in the y-axis.

The equation of the resulting curve is $\displaystyle y=\frac{2}{2x^2+2x+1}$

Determine the equation $\displaystyle y=f(x)$

What I did:

$\displaystyle -f(\frac{1}{2}x-1)$$\displaystyle =\frac{2}{2x^2+2x+1}$

2. Let $\displaystyle f_1(x)$, $\displaystyle f_2(x)$ and $\displaystyle f_3(x)$ be the results of transformations after the step A, B and C, respectively. Thus, $\displaystyle f_3(x) = 2/(2x^2+2x+1)$. Then $\displaystyle f_1(x) = f(x-1)$, $\displaystyle f_2(x)=f_1(2x)=f(2x-1)$ and $\displaystyle f_3(x)=f_2(-x)=f(-2x-1)$, not $\displaystyle -f(\frac{1}{2}x-1)$.

One way to find f(x) is assume that it has the form $\displaystyle 2/(Ax^2+Bx+C)$. Then $\displaystyle A(-2x-1)^2+B(-2x-1)+C=2x^2+2x+1$. By equating the corresponding coefficients, you can find A, B, and C.

Another, perhaps, more reliable, way is to apply the inverse transformations to $\displaystyle f_3(x)$.