Identify the focus of each. 31) y-6=(x-4)^2 32) (1)/(2)(x+2)=(y+9)^2 33) x^2+4x+4y+4=0 34) 2x^2-24x+y+70=0

1. Rewrite in standard parabola form <br /> Standard forms: Vertical: $(x-h)^2 = 4p(y-k)$, Horizontal: $(y-k)^2 = 4p(x-h)$. <br /><br />2. Problem 31 <br /> $y - 6 = (x - 4)^2 \ \Rightarrow \ (x - 4)^2 = y - 6$ → $4p = 1 \Rightarrow p = \frac14$. <br /> Vertex $(4, 6)$, vertical (opens up). Focus: $(4, 6 + \frac14) = (4, 6.25)$. <br /><br />3. Problem 32 <br /> $\frac12(x + 2) = (y + 9)^2 \ \Rightarrow \ (y + 9)^2 = \frac12(x + 2)$ → $4p = \frac12 \Rightarrow p = \frac18$. <br /> Vertex $(-2, -9)$, horizontal (opens right). Focus: $(-2 + \frac18, -9) = (-1.875, -9)$. <br /><br />4. Problem 33 <br /> $x^2 + 4x + 4y + 4 = 0 \ \Rightarrow \ x^2 + 4x = -4y - 4$. <br /> Complete square: $(x + 2)^2 - 4 = -4y - 4$ → $(x + 2)^2 = -4y$. <br /> $4p = -4 \Rightarrow p = -1$. Vertex $(-2, 0)$, opens down. Focus: $(-2, 0 + (-1)) = (-2, -1)$. <br /><br />5. Problem 34 <br /> $2x^2 - 24x + y + 70 = 0 \ \Rightarrow \ 2x^2 - 24x = -y - 70$. <br /> Factor $2$: $2(x^2 - 12x)$. Complete square: $x^2 - 12x + 36 = (x - 6)^2$. <br /> $2((x - 6)^2 - 36) = -y - 70$ → $2(x - 6)^2 - 72 = -y - 70$ → $2(x - 6)^2 - 2 = -y$. <br /> $2(x - 6)^2 + (-2) = -y$ → $(x - 6)^2 = -\frac12(y + 2)$. <br /> $4p = -\frac12 \Rightarrow p = -\frac18$. Vertex $(6, -2)$, opens down. Focus: $(6, -2 - \frac18) = (6, -2.125)$.