MHB Determine the equation of the parabola

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To determine the equation of the parabola with a range of y|y≥-6 and x-intercepts at -5 and 3, the equation is expressed as y = k(x+5)(x-3), where k is a positive constant. The vertex is located at the midpoint of the x-intercepts, which is at x = -1. The y-coordinate of the vertex is -6, leading to the vertex coordinates of (-1, -6). By substituting these coordinates into the equation, the value of k can be calculated. The final equation of the parabola can then be established based on these findings.
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Determine the equation of the parabola with range y|y≧-6 and x-intercepts at -5 and 3.
 
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$y = k(x+5)(x-3)$, where $k$ is a constant

$y \ge -6 \implies k > 0$

vertex is located at $(x,-6)$ where $x$ is midway between the two x-intercepts

determine the x-value of the vertex, then use the vertex coordinates to determine the value of $k$
 
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