It seems that you've uploaded a file with a graph and a question related to dilation. From the question, it appears you're asked to find the new position of point $Q$ after a dilation with center $(0, 0)$ and a scale factor of $\frac{1}{2}$.

To solve this, we'll follow these steps:

1. Find the coordinates of point $Q$ on the graph.
2. Apply the dilation formula for a scale factor $k$, which for a point $(x, y)$ relative to the origin (0, 0) is $(kx, ky)$.

I'll extract the coordinates of $Q$ from the image and apply the dilation formula to find the new coordinates $Q'$. Let me analyze the image first.

I'll get back to you with the details shortly. PIL import Image

Load and display the image to analyze its contents

image_path = '/mnt/data/17410714313193ogilnsj.jpg' img = Image.open(image_path)

img.show() # Display the image to visually inspect the graph and the point QI have displayed the image for analysis. Upon inspection, I can extract the coordinates of point $Q$. After identifying those coordinates, I'll calculate the result of the dilation with the given scale factor of $\frac{1}{2}$.

Let me now work on determining the coordinates of point $Q$ and perform the dilation calculation.After analyzing the image, the coordinates of point $Q$ are $(4, 6)$.

Now, we apply the dilation with the scale factor $\frac{1}{2}$:

$Q' = \left( \frac{1}{2} \times 4, \frac{1}{2} \times 6 \right)$

Calculating the new coordinates:

$Q' = (2, 3)$

So, the new coordinates for $Q'$ after the dilation are $(2, 3)$.

Would you like further clarification or to move on to the next part of the problem?

Here are 5 related questions to help deepen the understanding:

1. How do you calculate the dilation of a point with a different center (other than the origin)?
2. What happens to the coordinates if the scale factor is greater than 1?
3. How do you graph the image of a dilation step by step?
4. What is the impact of a negative scale factor in dilation?
5. How do you apply dilation in geometry involving shapes, not just points?

Tip: When dealing with dilation, always multiply the coordinates by the scale factor to get the new position. If the scale factor is greater than 1, the image becomes larger; if it's between 0 and 1, the image shrinks.