Question
Describe the transformation of the graph of the parent quadratic function to the graph of $f\left(x\right)=3\left(x+2\right)^2+1$
Answers
GPT-5 mini
Starting from the parent y = x^2:
1. Replace x by x + 2: shift left 2 units → (x+2)^2 (vertex moves from (0,0) to (-2,0)).
2. Multiply by 3: vertical stretch by factor 3 (makes the parabola narrower) → 3(x+2)^2 (y-values triple, vertex still at (-2,0)).
3. Add 1: shift up 1 unit → 3(x+2)^2 + 1 (vertex moves to (-2,1)).
So f(x) is an upward-opening parabola, vertically stretched by 3, shifted left 2 and up 1. Vertex: (-2, 1). Axis of symmetry: x = -2.
1. Replace x by x + 2: shift left 2 units → (x+2)^2 (vertex moves from (0,0) to (-2,0)).
2. Multiply by 3: vertical stretch by factor 3 (makes the parabola narrower) → 3(x+2)^2 (y-values triple, vertex still at (-2,0)).
3. Add 1: shift up 1 unit → 3(x+2)^2 + 1 (vertex moves to (-2,1)).
So f(x) is an upward-opening parabola, vertically stretched by 3, shifted left 2 and up 1. Vertex: (-2, 1). Axis of symmetry: x = -2.