compare the graphs of y= x squared and y = 2 and unknown squared number. How are they simialr? how are they differnt? use mathamatical terminolgy to support your answer. Make at least 3 comparisons between 2  functions 

Let's compare the graphs of the functions \( y = x^2 \) and \( y = 2 \) (an unknown squared number):

### 1. **Shape of the Graphs:**
   - **\( y = x^2 \):** This is a **parabola** that opens upwards. The graph is a U-shaped curve with its vertex at the origin \((0,0)\).
   - **\( y = 2 \) (unknown squared number):** This is a **horizontal line**. It represents a constant function where \( y \) is always equal to 2, regardless of the value of \( x \).

   **Comparison:** While the parabola is a curved line, the line \( y = 2 \) is straight. This means the shape of the graphs differs significantly—one is a curve, and the other is a flat line.

### 2. **Domain:**
   - **\( y = x^2 \):** The domain of \( y = x^2 \) is **all real numbers** \((-\infty, \infty)\). This means \( x \) can take any real value.
   - **\( y = 2 \) (unknown squared number):** The domain of \( y = 2 \) is also **all real numbers** \((-\infty, \infty)\).

   **Comparison:** Both functions have the same domain, which is all real numbers. This means you can input any \( x \)-value into either function.

### 3. **Range:**
   - **\( y = x^2 \):** The range of \( y = x^2 \) is \([0, \infty)\), meaning \( y \) takes on values from 0 to infinity. The graph never goes below \( y = 0 \).
   - **\( y = 2 \) (unknown squared number):** The range of \( y = 2 \) is exactly \(\{2\}\). The only value \( y \) can take is 2.

   **Comparison:** The range of the parabola is all non-negative real numbers starting from 0, whereas the range of the horizontal line is just a single value, 2. This is a significant difference in how the graphs behave vertically.

### Summary:
- **Shape:** Parabola vs. Horizontal Line.
- **Domain:** Both have the same domain of all real numbers.
- **Range:** \( y = x^2 \) has a range starting from 0 to infinity, while \( y = 2 \) has a fixed range of 2.

### Would you like more details or have any questions?

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Here are 5 related questions you might consider:
1. What would the graph look like for \( y = x^2 + 2 \)?
2. How would the graph of \( y = -x^2 \) compare to \( y = x^2 \)?
3. What are the intersections between \( y = x^2 \) and \( y = 2 \)?
4. How would the graph of \( y = x^2 + 2 \) compare to \( y = 2 \)?
5. What happens to the graph of \( y = x^2 \) if we change it to \( y = (x - 2)^2 \)?

**Tip:** When analyzing the graph of a function, always consider the shape, domain, and range to understand its behavior better.

Explore the similarities and differences between the graphs of y = x^2 and y = 2. Understand the mathematical concepts of parabolas and constant functions. Learn about their domain, range, and graphical representations. Suitable for students in Grades 9-12.

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