https://file.aiquickdraw.com/mathbot/file/1725899513882kt0l5wxb.jpg

Let's analyze the image to solve the questions based on the provided graph:

### For the graph, we can observe the following details:
- The graph represents a curve, likely a quadratic function.
- The \( x \)-axis is marked at intervals of 1, and the \( y \)-axis is also marked.
- The curve reaches its lowest point at \( (0, -5) \) and extends upwards on both sides.

### Answers to the questions:

**a) ¿Cuál es la imagen de -6?**  
To find the image of -6, locate \( x = -6 \) on the graph and see where the curve passes through this value of \( x \). The graph shows that at \( x = -6 \), \( y = 4 \).  
**Answer: \( 4 \)**.

**b) ¿Cuál es la imagen de 6?**  
For \( x = 6 \), locate this point on the graph. The graph shows that at \( x = 6 \), \( y = 4 \).  
**Answer: \( 4 \)**.

**c) ¿Cuál es la preimagen de -3?**  
To find the preimage of \( y = -3 \), check where the graph reaches \( y = -3 \). This happens at two points: \( x = -2 \) and \( x = 2 \).  
**Answer: \( -2 \) and \( 2 \)**.

**d) ¿Cuál es la preimagen de 2?**  
To find the preimage of \( y = 2 \), we look for the \( x \)-values where \( y = 2 \). According to the graph, this happens at \( x = -4 \) and \( x = 4 \).  
**Answer: \( -4 \) and \( 4 \)**.

**e) ¿En qué valores de \( x \) la función vale 0?**  
The function equals zero where the graph intersects the \( x \)-axis. This occurs at \( x = -5 \) and \( x = 5 \).  
**Answer: \( -5 \) and \( 5 \)**.

**f) ¿En qué valor de \( y \) el valor de \( x \) es 0?**  
When \( x = 0 \), the graph shows that \( y = -5 \).  
**Answer: \( -5 \)**.

**g) Dar dos valores de \( x \) con la misma imagen.**  
From the graph, we can see that \( x = -6 \) and \( x = 6 \) both have the image \( y = 4 \).  
**Answer: \( -6 \) and \( 6 \)**.

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Would you like further explanation on any specific part of the graph or function?

Here are five related questions for further exploration:
1. How can we determine the vertex of the graph?
2. What type of function might this graph represent (e.g., quadratic, cubic)?
3. How would you calculate the axis of symmetry from the graph?
4. What is the domain of the function represented in the graph?
5. How do we determine the range of the function?

**Tip:** To find the preimage of a function, you're looking for the \( x \)-values that correspond to a given \( y \)-value (i.e., where the curve hits a particular horizontal line).

Activity 4: Observing the graph and responding to questions related to images and preimages of a function.

Analyze a graph to determine the image and preimage values for given x and y coordinates. The problem explores fundamental concepts like finding where a function equals zero, and identifying the corresponding images and preimages for specific x and y values. Suitable for Grades 7-9.

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