To find the image of the points $-1$, $0$, and $1$ under the given transformation, we can use the provided formula:

$\frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{z - z_1}{z - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}$

Here, let's assume:

• $z_1 = -1$, $z_2 = 0$, $z_3 = 1$
• $w_1 = -i$, $w_2 = -1$, $w_3 = i$

We need to find the corresponding image $w$ for each $z$.

Step 1: Set up the equation for each $z$

1. For $z = -1$:

   Substitute $z = z_1 = -1$ into the equation: $\frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{-1 - z_1}{-1 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}$

   Simplifying: $\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{-1 + 1}{-1 - 1} \cdot \frac{0 - 1}{0 + 1}$

2. For $z = 0$:

   Substitute $z = z_2 = 0$ into the equation: $\frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{0 - z_1}{0 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}$

   Simplifying: $\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{1}{-1} \cdot \frac{-1}{1}$

3. For $z = 1$:

   Substitute $z = z_3 = 1$ into the equation: $\frac{w - w_1}{w - w_3} \cdot \frac{w_2 - w_3}{w_2 - w_1} = \frac{1 - z_1}{1 - z_3} \cdot \frac{z_2 - z_3}{z_2 - z_1}$

   Simplifying: $\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{2}{0} \cdot \frac{-1}{2}$

Step 2: Solve each equation for $w$

Given that the transformation must map $-1$ to $-i$, $0$ to $-1$, and $1$ to $i$, solve the resulting simplified equations for $w$.

For $z = -1$, solve:

$\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = \frac{0}{-2} \cdot \frac{-1}{2}$ This equation simplifies to $w = -i$.

For $z = 0$, solve:

$\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = 1$ This equation simplifies to $w = -1$.

For $z = 1$, solve:

$\frac{w + i}{w - i} \cdot \frac{-1 - i}{-1 + i} = -\infty$ This equation simplifies to $w = i$.

So the image points are:

• $z = -1$ maps to $w = -i$
• $z = 0$ maps to $w = -1$
• $z = 1$ maps to $w = i$

Would you like to see more details about these transformations, or do you have any questions? Here are five related questions to consider:

1. How do you solve for $w$ in each transformation equation?
2. What is the significance of each term in the transformation formula?
3. How would the transformation change if the points $w_1$, $w_2$, and $w_3$ were different?
4. Can this transformation be generalized to map other sets of points?
5. What type of transformation is this, geometrically?

Tip: Always check the consistency of each step when applying complex transformations, as errors can easily propagate.