The problem in the image asks to determine the images of given points after dilation (scaling) from the origin $O$ with the specified scaling factor. Let's solve the problems step by step.

Formula for Dilation

When dilating a point $P(x, y)$ from the origin $O$, the new point $P'(x', y')$ can be found by multiplying both the x-coordinate and the y-coordinate by the scaling factor $k$: $(x', y') = (k \cdot x, k \cdot y)$ where $k$ is the scaling factor.

Let's solve each part:

a. Point $(3, 4)$ with scaling factor $2$: $(3', 4') = (2 \cdot 3, 2 \cdot 4) = (6, 8)$ So, the new point is $(6, 8)$.

b. Point $(-2, 4)$ with scaling factor $-4$: $(-2', 4') = (-4 \cdot (-2), -4 \cdot 4) = (8, -16)$ So, the new point is $(8, -16)$.

c. Point $(2, -1)$ with scaling factor $3$: $(2', -1') = (3 \cdot 2, 3 \cdot (-1)) = (6, -3)$ So, the new point is $(6, -3)$.

d. Point $(-6, 2)$ with scaling factor $\frac{1}{2}$: $(-6', 2') = \left(\frac{1}{2} \cdot (-6), \frac{1}{2} \cdot 2\right) = (-3, 1)$ So, the new point is $(-3, 1)$.

e. Point $(12, -27)$ with scaling factor $\frac{1}{3}$: $(12', -27') = \left(\frac{1}{3} \cdot 12, \frac{1}{3} \cdot (-27)\right) = (4, -9)$ So, the new point is $(4, -9)$.

Final Answers:

• a. $(6, 8)$
• b. $(8, -16)$
• c. $(6, -3)$
• d. $(-3, 1)$
• e. $(4, -9)$

Would you like any further details on dilation or have any questions?

Related questions:

1. How do you perform a dilation if the center of dilation is not the origin?
2. What happens to the orientation of a shape when dilated with a negative scaling factor?
3. Can dilation result in a smaller image? If yes, under what conditions?
4. How does dilation affect the distances between points on a geometric figure?
5. What are some real-world applications of dilation in mathematics?

Tip: When the scaling factor is less than 1, the image becomes smaller, while a factor greater than 1 enlarges the image.