F.5 Function & Graphs

Note that the roots of F(x) = 0 are 1, -2, -1/2.
And the roots of G(x) = 0 are -1, 2, -4.

The roots are doubled from F(x) = 0 to G(x) = 0.
1 × 2 = 2
-2 × 2 = -4
-1/2 × 2 = -1

Suppose F(a) = 0, then G(2a) = 0.

First consider
G(2x) = (2x + 1)(2x - 2)(2x + 4)
　　 = (2x + 1)(2)(x - 1)(2)(x + 2)
　　 = 4(2x + 1)(x - 1)(x + 2)
　　 = 4F(x)

Let y = 2x, x = y/2.
G(y) = 4F(y/2)

Therefore,
G(x) = 4F(x/2).

That means, y = F(x) is transformed to y = G(x) in the following ways:
★ First, horizonally scale the graph 2 times (about the y-axis) as before.
★ Then, vertically scale the graph 4 times (about the x-axis) as before.