Question

explain the translations of this quadratic function using the key features: $y=(x-3)^2+2$
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Explanation:

Start with the parent quadratic function $y=x^2$, which has its vertex at $(0,0)$.

1. The term $(x-3)$ inside the square shifts the graph horizontally: subtracting 3 from $x$ moves the vertex 3 units to the right.
2. The $+2$ outside the square shifts the graph vertically: adding 2 moves the vertex 2 units upward.

The vertex of the transformed function $y=(x-3)^2+2$ is at $(3,2)$, reflecting these two translations.

Answer:

The quadratic function $y=(x-3)^2+2$ is a translation of the parent function $y=x^2$: it is shifted 3 units to the right and 2 units upward, with its vertex moving from $(0,0)$ to $(3,2)$.