Question

which translation maps the vertex of the graph of the function $f(x) = x^2$ onto the vertex of the function $g(x) = x^2 + 2x +1$?

• right 1 unit
• left 1 unit
• right 2 units
• left 2 units

Explanation:

Step1: Find vertex of \( f(x) = x^2 \)

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( f(x) = x^2 \), it can be written as \( f(x) = 1(x - 0)^2 + 0 \), so the vertex is \((0, 0)\).

Step2: Rewrite \( g(x) = x^2 + 2x + 1 \) in vertex form

Factor \( g(x) \): \( g(x) = (x + 1)^2 \). In vertex form \( y = a(x - h)^2 + k \), here \( h = -1 \), \( k = 0 \), so the vertex is \((-1, 0)\).

Step3: Determine translation

To go from \((0, 0)\) to \((-1, 0)\), we move left 1 unit (since moving from \( x = 0 \) to \( x = -1 \) is a leftward shift of 1 unit).

Answer:

B. left 1 unit