To find the invariant point under the combined transformation, follow these steps:

Step 1: Define the Transformations

1. Rotation of 180° about (-2,0)
   The formula for rotating a point $(x, y)$ by 180° about $(h, k)$ is:
   $(x', y') = (2h - x, 2k - y)$ Substituting $h = -2$ and $k = 0$, we get:
   $(x', y') = (-4 - x, -y)$

2. Translation by $\begin{pmatrix} -3 \\ 2 \end{pmatrix}$
   This means shifting the coordinates as:
   $(x'', y'') = (x' - 3, y' + 2)$

Step 2: Find the Invariant Point

An invariant point remains unchanged after both transformations. That means:
$(x, y) = (x'', y'')$ Substituting the transformations:

$(x, y) = (-4 - x - 3, -y + 2)$

This gives the equations:

1. $x = -4 - x - 3$
   $x + x = -7$ $2x = -7 \Rightarrow x = -\frac{7}{2}$

2. $y = -y + 2$
   $y + y = 2$ $2y = 2 \Rightarrow y = 1$

Final Answer:

The invariant point is $\left(-\frac{7}{2}, 1 \right)$.

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Do you want more details or have any questions?

Here are some related questions:

1. How do you find the center of rotation for a given transformation?
2. What are the properties of a 180° rotation transformation?
3. How do you determine if a transformation has an invariant point?
4. What are other transformations that might leave a point unchanged?
5. How does a 90° rotation differ from a 180° rotation?

Tip:

Always check for invariant points in reflection, rotation, and translation by solving for points that satisfy the equation $(x, y) = (x'', y'')$.