Answer to Question #297202 in Analytic Geometry for Victoria

Question

The coordinates of the image of 𝑃(𝑥, 𝑦) when 𝑦 = 𝑓(𝑥) is transformed to

𝑦 = 2 𝑓(𝑥 − 3) − 1 are 𝑃′

(2, 3). Find the original point (𝑥, 𝑦).

Answer 1

Let the original point be $P(𝑥, 𝑦).$

$$f(x-3): P_1(x+3, y)$$

$$2f(x-3): P_2(x+3, 2y)$$

$$2f(x-3)-1: P_1(x+3, 2y-1)$$

Then

$$x+3=2=>x=-1$$

$$2y-1=3=>y=2$$

The original point is $P(-1, 2).$