To find an equation of a parabola with a given vertex, we can use the vertex form of a quadratic equation, which is:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola and \(a\) determines the direction and width of the parabola.
Given the vertex \((-4, 5)\), we can substitute \(h = -4\) and \(k = 5\) into the equation:
\[ y = a(x + 4)^2 + 5 \]
At this point, \(a\) can be any non-zero constant:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
For example:
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If we choose \(a = 1\), the equation becomes: \[ y = (x + 4)^2 + 5 \]
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If we choose \(a = -1\), the equation becomes: \[ y = -(x + 4)^2 + 5 \]
You can choose any value for \(a\) to create various equations with the vertex \((-4, 5)\).