Let's go through each problem step-by-step, showing the full work for how they are solved.

Problem 41:

Vertex: $(1, -2)$, Point: $(-1, 14)$

The equation of a parabola in vertex form is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. Here, the vertex is $(1, -2)$, so we substitute $h = 1$ and $k = -2$ into the equation: $y = a(x - 1)^2 - 2$ Now, use the point $(-1, 14)$ to find $a$. Substituting $x = -1$ and $y = 14$ into the equation: $14 = a(-1 - 1)^2 - 2$ Simplifying: $14 = a(-2)^2 - 2$ $14 = 4a - 2$ Add 2 to both sides: $16 = 4a$ Solve for $a$: $a = 4$ Thus, the equation of the parabola is: $y = 4(x - 1)^2 - 2$

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Problem 42:

Vertex: $(-4, -1)$, Point: $(-2, -4)$

The equation of the parabola is: $y = a(x - h)^2 + k$ Substitute $h = -4$ and $k = -1$: $y = a(x + 4)^2 - 1$ Use the point $(-2, -4)$ to solve for $a$: $-4 = a(-2 + 4)^2 - 1$ Simplifying: $-4 = a(2)^2 - 1$ $-4 = 4a - 1$ Add 1 to both sides: $-3 = 4a$ Solve for $a$: $a = -\frac{3}{4}$ Thus, the equation of the parabola is: $y = -\frac{3}{4}(x + 4)^2 - 1$

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Problem 43:

Vertex: $\left(\frac{1}{2}, 1\right)$, Point: (\left(-2, \frac{21}{