1. **Problem Statement:** We have a shape A (a right triangle) with vertices approximately at $(-4,4)$, $(-4,2)$, and $(-3,2)$ on the coordinate plane. (a) We need to find the image of shape A after a translation by the vector $(8, -6)$. (b) We need to find the image of shape A after a reflection in the line $y = -1$. (c) Describe the single transformation that maps shape A onto shape B. (d) Describe the single transformation that maps shape A onto shape C. --- 2. **Formulas and Rules:** - Translation by vector $(a,b)$ moves every point $(x,y)$ to $(x+a, y+b)$. - Reflection in the line $y = k$ maps a point $(x,y)$ to $(x, 2k - y)$. - To describe a transformation mapping one shape to another, identify the type (translation, reflection, rotation, dilation) and parameters. --- 3. **Step-by-step Solutions:** **(a) Translation by $(8, -6)$:** - Original vertices of A: $(-4,4)$, $(-4,2)$, $(-3,2)$. - Apply translation: $(x,y) \to (x+8, y-6)$. - New vertices: - $(-4+8, 4-6) = (4, -2)$ - $(-4+8, 2-6) = (4, -4)$ - $(-3+8, 2-6) = (5, -4)$ So, the image of A after translation is a triangle with vertices at $(4,-2)$, $(4,-4)$, and $(5,-4)$. **(b) Reflection in the line $y = -1$:** - Reflection formula: $(x,y) \to (x, 2(-1) - y) = (x, -2 - y)$. - Apply to vertices: - $(-4,4) \to (-4, -2 - 4) = (-4, -6)$ - $(-4,2) \to (-4, -2 - 2) = (-4, -4)$ - $(-3,2) \to (-3, -2 - 2) = (-3, -4)$ The reflected image is a triangle with vertices at $(-4,-6)$, $(-4,-4)$, and $(-3,-4)$. **(c) Transformation mapping A onto B:** - Shape B is near $(-1,-2)$ and below, which matches the translation of A by vector $(3, -4)$ approximately. - Check translation vector from A to B: - From $(-4,4)$ to $(-1,-2)$ is $(3, -6)$. - From $(-4,2)$ to $(-1,-4)$ is $(3, -6)$. - From $(-3,2)$ to $(0,-4)$ is $(3, -6)$. So, the single transformation is a translation by vector $(3, -6)$. **(d) Transformation mapping A onto C:** - Shape C is a line segment in quadrant I between approximately $(2,5)$ and $(5,6)$. - Since A is a triangle and C is a line segment, the transformation is likely a rotation or reflection combined with dilation or projection. - Observing the points, C looks like a rotated and translated image of A. - The transformation is a rotation of 90 degrees clockwise about the origin followed by a translation. --- **Final answers:** (a) Translation by vector $(8, -6)$. (b) Reflection in the line $y = -1$. (c) Translation by vector $(3, -6)$. (d) Rotation 90° clockwise about the origin followed by translation to position C.