Let's go through the problem step-by-step, interpreting the tasks based on the details provided.

Problem Summary

The exercise involves transformations applied to a triangle with vertices:

• $A = (-2, 0)$
• $B = (1, 2)$
• $C = (0, 1)$

Several transformations are performed on this triangle, labeled as:

• $A'B'C'$: the image of $ABC$ after a transformation.
• $A''B''C''$: the image of $A'B'C'$ under another transformation.

The tasks are as follows:

Questions Analysis

1. Plot the three triangles $ABC$, $A'B'C'$, and $A''B''C''$

   • This involves plotting the original triangle and the images after each transformation.

2. Find a single matrix that maps $A''B''C''$ onto $ABC$

   • You need to determine the composition of transformations that would reverse the effect of transforming $ABC$ to $A''B''C''$.

3. Find a single matrix that maps $A'B'C'$ onto $ABC$

   • This requires identifying a transformation that maps the first image, $A'B'C'$, back to the original triangle $ABC$.

4. If triangle $ABC$ is sheared with a shear factor of 2 along the x-axis, find the coordinates of the resulting image

   • Shear transformation should be applied to $ABC$ to get new coordinates after applying this shear.

Step-by-Step Approach

1. Plotting the triangles:

   • Identify the transformation matrices that produce $A'B'C'$ and $A''B''C''$ if given, or assume generic transformations (such as reflections, rotations, etc.) if not specified.
   • Apply these matrices to $ABC$'s vertices and plot the coordinates accordingly.

2. Matrix for Mapping $A''B''C''$ onto $ABC$:

   • If transformations from $ABC \to A'B'C'$ and $A'B'C' \to A''B''C''$ are known, find the inverse of their product to map $A''B''C''$ back to $ABC$.

3. Matrix for Mapping $A'B'C'$ onto $ABC$:

   • Similarly, find the inverse of the transformation that mapped $ABC$ to $A'B'C'$.

4. Shear Transformation with Factor 2:

   • Use the shear matrix for the x-axis shear:
     $\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$
   • Apply this matrix to each vertex of $ABC$ to get the new coordinates.

Would you like me to go through each calculation in detail, starting with plotting the transformations or focusing on a specific question?

Follow-Up Questions

1. How do we determine the exact transformation matrices for $A'B'C'$ and $A''B''C''$?
2. What are the coordinates of $A'B'C'$ and $A''B''C''$ after each transformation?
3. What is the procedure to find the inverse of a transformation matrix?
4. How does a shear transformation differ from rotation and reflection in terms of effect on a shape?
5. What is the significance of invariant lines in transformations?

Tip

When working with transformations, understanding how each matrix affects coordinates is essential. Start with simple transformations (like reflections and rotations) before moving to combinations to build intuition.