What part of the function $$ f(x)=a(x-h)^{2}+k $$ affects the range?

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1. The quadratic function in vertex form is given by $$ f(x)=a(x-h)^2+k. $$ 2. The vertex of the parabola is at the point $$(h, k)$$. 3. The parameter $$a$$ determines the direction of the opening of the parabola: - If $$a>0$$, the parabola opens upward and the vertex is the minimum point. - If $$a<0$$, the parabola opens downward and the vertex is the maximum point. 4. The constant $$k$$ represents the vertical shift and establishes the $$y$$-coordinate of the vertex. This $$y$$-coordinate serves as the boundary for the range: - For $$a>0$$, the range is $$\left[k, \infty\right)$$. - For $$a<0$$, the range is $$\left(-\infty, k\right]$$. 5. Therefore, the part of the function that directly affects the range is the constant $$k$$, along with the sign of $$a$$, which determines whether $$k$$ is a minimum or a maximum value. Thus, $$k$$ (in combination with the sign of $$a$$) affects the range. The constant $$k$$ in the function $$f(x)=a(x-h)^2+k$$ determines the range. If $$a$$ is positive, the range is from $$k$$ to infinity. If $$a$$ is negative, the range is from negative infinity to $$k$$.

What part of the function affects the range?

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