Suppose the image, h’, under translation of h(s + 2, r − 5) is (2s − 1, 3r + 7). 
What is the translation vector, < a, b >, of the transformation?


To find the translation vector \(\langle a, b \rangle\), we need to determine how the coordinates \( (s, r) \) are translated to \( (2s - 1, 3r + 7) \).

Given the transformation is:
\[
h'(s + 2, r - 5) = (2s - 1, 3r + 7),
\]
we can compare this with a typical translation, which is of the form:
\[
h'(s + a, r + b).
\]

Here, the expressions in the transformed coordinates correspond to changes in the original coordinates \( (s, r) \):

- \( s + 2 \) in the original becomes \( 2s - 1 \) in the transformed,
- \( r - 5 \) in the original becomes \( 3r + 7 \) in the transformed.

Now, let's set up the equations for the translation vector:

1. From \( s + 2 = 2s - 1 \), solve for \( a \):
   \[
   s + 2 = 2s - 1 \implies s - 2s = -1 - 2 \implies -s = -3 \implies s = 3.
   \]

2. From \( r - 5 = 3r + 7 \), solve for \( b \):
   \[
   r - 5 = 3r + 7 \implies r - 3r = 7 + 5 \implies -2r = 12 \implies r = -6.
   \]

Thus, the translation vector is:
\[
\langle a, b \rangle = \langle 3, -6 \rangle.
\]

Would you like further explanation or more details on these steps? Here are five related questions:

1. How do you find the general form of a translation matrix?
2. What role do translation vectors play in affine transformations?
3. How do translation vectors differ from rotation and scaling vectors?
4. How can translation be represented in a matrix form?
5. Can translation vectors be combined with other transformations?

**Tip:** When solving for transformations, always compare each component separately for clarity.

Suppose the image, h’, under translation of h(s + 2, r − 5) is (2s − 1, 3r + 7). What is the translation vector, < a, b >, of the transformation?

Learn how to find the translation vector < a, b > for a coordinate transformation from h(s + 2, r - 5) to (2s - 1, 3r + 7). This problem involves solving linear equations and applying transformation principles in coordinate geometry. Suitable for students in Grades 9-11.

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