Let's go step-by-step through the problem based on the image you provided:

1. Plot points A, B, and C on the grid and join them:

• A = (1, 1)
• B = (3, 3)
• C = (4, 1)

After plotting these points on the given grid, you can connect them to form triangle ABC.

2. Transformations on triangle ABC:

a) Translate by $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$:

• Translation involves shifting the triangle.
  • For point A: $(1, 1) \rightarrow (1 + 2, 1 + 1) = (3, 2)$
  • For point B: $(3, 3) \rightarrow (3 + 2, 3 + 1) = (5, 4)$
  • For point C: $(4, 1) \rightarrow (4 + 2, 1 + 1) = (6, 2)$
• New coordinates of triangle A'B'C' after translation are: A'(3, 2), B'(5, 4), C'(6, 2).

b) Reflect in the y-axis:

• Reflection in the y-axis changes the sign of the x-coordinate.
  • For point A: $(1, 1) \rightarrow (-1, 1)$
  • For point B: $(3, 3) \rightarrow (-3, 3)$
  • For point C: $(4, 1) \rightarrow (-4, 1)$
• New coordinates of triangle A"B"C" after reflection are: A"(-1, 1), B"(-3, 3), C"(-4, 1).

c) Rotate clockwise by 180° about (0, 0):

• Rotation by 180° changes both the x- and y-coordinates to their opposites.
  • For point A: $(1, 1) \rightarrow (-1, -1)$
  • For point B: $(3, 3) \rightarrow (-3, -3)$
  • For point C: $(4, 1) \rightarrow (-4, -1)$
• New coordinates of triangle A'''B'''C''' after 180° rotation are: A'''(-1, -1), B'''(-3, -3), C'''(-4, -1).

d) Reflect in the line $y = -1$:

• To reflect across the line $y = -1$, calculate the distance from each point to $y = -1$, then move each point the same distance in the opposite direction.
  • For point A: The distance from $(1, 1)$ to $y = -1$ is 2 units. So, after reflection, $A \rightarrow (1, -3)$.
  • For point B: The distance from $(3, 3)$ to $y = -1$ is 4 units. So, after reflection, $B \rightarrow (3, -5)$.
  • For point C: The distance from $(4, 1)$ to $y = -1$ is 2 units. So, after reflection, $C \rightarrow (4, -3)$.
• New coordinates of triangle A''''B''''C'''' after reflection are: A''''(1, -3), B''''(3, -5), C''''(4, -3).

Let me know if you'd like any further details on the steps or have additional questions.

Here are five relative questions:

1. How do you plot points on a coordinate plane?
2. What is the general rule for translating points in a coordinate plane?
3. What happens to the coordinates of a point when reflecting in the y-axis?
4. How do you calculate a 180° rotation of a point around the origin?
5. How can you determine the reflection of a point across a horizontal line like $y = -1$?

Tip:

When reflecting over a line, first calculate the perpendicular distance from the point to the line. Then move the same distance in the opposite direction!