https://file.aiquickdraw.com/mathbot/file/1726403401355tsxv6vhk.jpg

The problem in the image asks for the image of the point \( Q(-4, 5) \) after two transformations:

1. **Translation** by \( T = \left( \frac{-1}{4} \right) \), which seems to refer to a vector translation of \( \left( -1, 4 \right) \).
2. **Rotation** by \( 90^\circ \) counterclockwise about the origin \( O(0, 0) \).

### Step 1: Translation
We first translate point \( Q(-4, 5) \) by the vector \( (-1, 4) \).

- New coordinates of \( Q' \):
  \[
  Q' = (-4 - 1, 5 + 4) = (-5, 9)
  \]

### Step 2: Rotation
Next, we rotate the point \( Q'(-5, 9) \) counterclockwise by \( 90^\circ \) around the origin \( O(0, 0) \). The transformation formula for a \( 90^\circ \) counterclockwise rotation is:

- New coordinates of \( Q'' \) after rotation:
  \[
  Q'' = (-y, x)
  \]
  Applying this to \( (-5, 9) \):
  \[
  Q'' = (-9, -5)
  \]

### Final Answer:
The image of the point after translation and rotation is \( (-9, -5) \).

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Would you like further details or explanations? Here are some related questions to practice:

1. How would the result change if the rotation was \( 90^\circ \) clockwise?
2. What is the general formula for rotating a point by \( 180^\circ \)?
3. How do you apply a translation using different vectors?
4. What are the coordinates of the point if the rotation was around a different center?
5. What happens if the translation vector is reversed?

**Tip:** Always apply translation first, followed by rotation, unless the problem specifies otherwise.

Bayangan titik Q(-4,5) oleh translasi T = (-1,4) dilanjutkan dengan rotasi sejauh 90 derajat berlawanan arah jarum jam terhadap pusat O(0,0) adalah ...

Learn how to find the image of point Q(-4,5) after translation by T = (-1, 4) followed by a 90-degree counterclockwise rotation about the origin. This problem demonstrates key concepts in translation and rotation, suitable for students in Grades 8-10.

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