1. Problem: Find the images of points under given dilations. 2. Formula for dilation of a point $Q(x,y)$ about center $P(x_p,y_p)$ with scale factor $k$ is: $$Q' = (x_p + k(x - x_p), y_p + k(y - y_p))$$ If center is origin $[0,k]$, then: $$Q' = (kx, ky)$$ 3. Calculate each: a. $A(3,5)$ by dilation about origin with scale factor 4: $$A' = (4 \times 3, 4 \times 5) = (12, 20)$$ b. $B(-4,2)$ by dilation about origin with scale factor -3: $$B' = (-3 \times -4, -3 \times 2) = (12, -6)$$ c. $C(-5,-1)$ by dilation about $P(2,3)$ with scale factor 2: $$C'_x = 2 + 2(-5 - 2) = 2 + 2(-7) = 2 - 14 = -12$$ $$C'_y = 3 + 2(-1 - 3) = 3 + 2(-4) = 3 - 8 = -5$$ So, $C' = (-12, -5)$ d. $D(\frac{1}{2}, -2)$ by dilation about $P(4,4)$ with scale factor $-\frac{1}{4}$: $$D'_x = 4 + (-\frac{1}{4})(\frac{1}{2} - 4) = 4 - \frac{1}{4} \times \frac{7}{2} = 4 - \frac{7}{8} = \frac{32}{8} - \frac{7}{8} = \frac{25}{8}$$ $$D'_y = 4 + (-\frac{1}{4})(-2 - 4) = 4 - \frac{1}{4} \times (-6) = 4 + \frac{6}{4} = 4 + \frac{3}{2} = \frac{8}{2} + \frac{3}{2} = \frac{11}{2}$$ So, $D' = (\frac{25}{8}, \frac{11}{2})$ e. $E(2,6)$ by dilation about $P(5,-1)$ with scale factor $-\frac{1}{2}$: $$E'_x = 5 + (-\frac{1}{2})(2 - 5) = 5 - \frac{1}{2} \times (-3) = 5 + \frac{3}{2} = \frac{10}{2} + \frac{3}{2} = \frac{13}{2}$$ $$E'_y = -1 + (-\frac{1}{2})(6 + 1) = -1 - \frac{1}{2} \times 7 = -1 - \frac{7}{2} = -\frac{2}{2} - \frac{7}{2} = -\frac{9}{2}$$ So, $E' = (\frac{13}{2}, -\frac{9}{2})$ Final answers for question 4: - $A' = (12, 20)$ - $B' = (12, -6)$ - $C' = (-12, -5)$ - $D' = (\frac{25}{8}, \frac{11}{2})$ - $E' = (\frac{13}{2}, -\frac{9}{2})$