https://file.aiquickdraw.com/mathbot/file/1727101571757nbd7nejz.jpg

To help you graph the equations provided in the image:

1. **Equation 1: \( y = x^2 + 3 \)**
   - This represents a parabola, shifted upward by 3 units from the standard parabola \( y = x^2 \).
   - The vertex is at \( (0, 3) \).
   - It opens upward, as the coefficient of \( x^2 \) is positive.

2. **Equation 2: \( y = x \)**
   - This is a straight line with a slope of 1, passing through the origin.
   - The line will pass through points like \( (0,0) \), \( (1,1) \), \( (-1,-1) \), and so on.

### Steps to Graph:
1. **Graph \( y = x^2 + 3 \)**: Start by plotting the vertex at \( (0, 3) \) and then plot other points, such as:
   - \( (-2, 7) \), \( (-1, 4) \), \( (1, 4) \), \( (2, 7) \), etc.
2. **Graph \( y = x \)**: This is a diagonal line with points like \( (0, 0) \), \( (1, 1) \), and \( (-1, -1) \).

The graphs will intersect at certain points. You can identify these intersections by solving the system of equations \( y = x^2 + 3 \) and \( y = x \) simultaneously.

Would you like to go over the process of finding the points of intersection in detail?

Here are 5 questions to extend your understanding:
1. What happens when you solve \( y = x^2 + 3 \) and \( y = x \) simultaneously?
2. How would the graph change if the parabola were \( y = -x^2 + 3 \)?
3. What is the significance of the slope of the line \( y = x \)?
4. Can you predict where the two graphs intersect by estimation?
5. What happens to the graph of \( y = x^2 + 3 \) if the "+3" changes to another number?

**Tip:** Always check the direction of a parabola by the sign of the quadratic term—positive opens upwards, and negative opens downwards!

Graph the equations y = x^2 + 3 and y = x.

Learn how to graph the quadratic function y = x^2 + 3 and the linear function y = x. This problem demonstrates important concepts in graphing, including the properties of parabolas and straight lines. Suitable for students in Grades 9-10.

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