- Identify original triangle vertices: \(A(-4, 1)\), \(B(5, 2)\), \(C(3, 2)\).
- Apply the transformation \((x, y) \rightarrow (2x, y)\) to find the new vertices: \(A'(-8, 1)\), \(B'(10, 2)\), \(C'(6, 2)\).
- Evaluate which given option is not in the set of new vertices.
- The point that is not a vertex is \(\boxed{4)\ (-8, 2)}\).
Explanation
- Analyze the initial data and the transformation.
The original triangle is defined by the vertices \(A = (-4, 1)\), \(B = (5, 2)\), and \(C = (3, 2)\). We are applying a horizontal stretch transformation with a scale factor of \(2\) with respect to the \(y\)-axis (\(x=0\)). The transformation rule is given by \((x, y) \rightarrow (2x, y)\). This rule means that every x-coordinate is multiplied by 2, while the y-coordinate remains unchanged.
- Calculate the new triangle vertices.
To find the vertices of the new triangle, we apply the transformation \((x, y) \rightarrow (2x, y)\) to each of the original vertices:
- For vertex \(A(-4, 1)\), the transformed vertex is \(A' = (2 \times -4, 1) = (-8, 1)\).
- For vertex \(B(5, 2)\), the transformed vertex is \(B' = (2 \times 5, 2) = (10, 2)\).
- For vertex \(C(3, 2)\), the transformed vertex is \(C' = (2 \times 3, 2) = (6, 2)\).
Thus, the set of vertices of the new triangle is \(\{(-8, 1), (10, 2), (6, 2)\}\).
- Identify the point that is not a vertex.
We compare the list of new vertices \(\{(-8, 1), (10, 2), (6, 2)\}\) with the given options:
1) \((-8, 1)\) - This is a vertex of the new triangle.
2) \((10, 2)\) - This is a vertex of the new triangle.
3) \((6, 2)\) - This is a vertex of the new triangle.
4) \((-8, 2)\) - This point is not present in our set of new vertices.
Therefore, the point \((-8, 2)\) is not a vertex of the transformed triangle. The final answer is option 4.
Examples
Horizontal stretches are common in graphic design and image processing. For example, when you resize a picture by stretching it horizontally to fit a wider screen, you are essentially applying a transformation similar to \((x, y) \rightarrow (kx, y)\). If you know the coordinates of key features in the image, this calculation helps you predict their new positions on the screen after the stretch. Understanding how these transformations affect shapes is a foundational skill in geometry and computer graphics.