Question

4. F-BF.2.3

The vertex of the parabola represented by
f(x) = x2 - 4x + 3 has coordinates (2, -1).
Find the coordinates of the vertex of the
parabola defined by g(x) = f(x - 2).
Explain how you arrived at your answer.

Answer 1

(4;-1).

Explanation:

1) to write new equation:

f(x-2)=(x-2)²-4(x-2)+3; ⇒ f(x-2)=x²-8x+15;

2) to define the new vertex:

(4; -1).

PS. the suggested way is not the shortest one; it's just to move the vertex to the right direction on 2 units.

Answer 2

Vertex of g(x) is (4, -1)

Explanation:

Method 1

Using translations rules:

`\textsf{when }\:a > 0: \quad f(x-a) \implies f(x) \: \textsf{translated}\:a\:\textsf{units right}`

`\textsf{Therefore }\quad f(x-2) \implies f(x) \: \textsf{translated}\:2\:\textsf{units right}`

If the vertex of f(x) is (2, -1)

then the vertex of g(x) is (2 + 2, -1) = (4, -1)

Method 2

Vertex form of a parabola

`y=a(x-h)^2+k`

(where (h, k) is the vertex and a is some constant)

Given function:
`f(x)=x^2-4x+3`

If the vertex of f(x) is (2, -1), then f(x) written in vertex form is:

`f(x)=(x-2)^2-1`

`\begin{aligned} \textsf{If}\quad g(x) & =f(x-2)\\\implies g(x) &=((x-2)-2)^2-1\\&=(x-4)^2-1\end{aligned}`

Therefore, the vertex of g(x) is (4, -1)