Use the function rules to find the coordinates of triangle A " B" C" and predict the quadrant it will be in after the transformations. Triangles ABC, with coordinates A (-8,4), B (-5,8), and C (-3,3), is transformed by (x,y) > (x+ 5, y -3). Create the rule composition transformation and predict what quadrant the image will be in.

Answer 1

To apply the transformation to triangle ABC, we start by using the given rule for the transformation, which is \((x, y) \rightarrow (x + 5, y - 3)\). This means we will add 5 to the \(x\)-coordinate and subtract 3 from the \(y\)-coordinate of each vertex of triangle ABC.

We will apply this transformation to each vertex \(A\), \(B\), and \(C\):

1. For point \(A(-8, 4)\): \[ A' = (-8 + 5, 4 - 3) = (-3, 1) \]

2. For point \(B(-5, 8)\): \[ B' = (-5 + 5, 8 - 3) = (0, 5) \]

3. For point \(C(-3, 3)\): \[ C' = (-3 + 5, 3 - 3) = (2, 0) \]

Now, the coordinates of the transformed triangle \(A'B'C'\) are:

• \(A'(-3, 1)\)
• \(B'(0, 5)\)
• \(C'(2, 0)\)

Next, we need to determine the quadrant in which these new points are located:

• Point \(A'(-3, 1)\): \(x < 0\) and \(y > 0\) → Quadrant II
• Point \(B'(0, 5)\): \(x = 0\) and \(y > 0\) → This point lies on the positive \(y\)-axis.
• Point \(C'(2, 0)\): \(x > 0\) and \(y = 0\) → This point lies on the positive \(x\)-axis.

To summarize:

• The vertex \(A'\) is in Quadrant II.
• The vertex \(B'\) is on the positive \(y\)-axis.
• The vertex \(C'\) is on the positive \(x\)-axis.

Overall, the majority of the transformed triangle's vertices are in Quadrant II or on the axes. The triangle itself will primarily be situated around the area defined by Quadrant II and the axes.

Prediction

Thus, after the transformation, triangle \(A'B'C'\) will be primarily located around Quadrant II and the axes, but not specifically confined to one quadrant.