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This image shows a coordinate plane with a parabola and a line. The parabola is represented by the equation \( y = x^2 \), and the line is represented by the equation \( y = x + 4 \). There are also several numbered regions (1 through 7) marked in the diagram. Here's what the elements represent:

1. **The Parabola \( y = x^2 \)**: This curve is symmetric about the y-axis and represents a quadratic function. The vertex of the parabola is at the origin (0, 0).

2. **The Line \( y = x + 4 \)**: This line has a slope of 1 and a y-intercept of 4. It is an upward-sloping straight line that intersects the parabola.

3. **The Intersection of the Parabola and Line**: The points where the line and the parabola intersect are of interest, which can be found by solving the system of equations \( y = x^2 \) and \( y = x + 4 \).

4. **Regions and Shading**: The numbered regions (e.g., Region 1, Region 2) indicate the areas of the graph that are divided by the parabola and line. For instance:
   - Region 1 lies below both the parabola and the line.
   - Region 5 lies between the parabola and the line.
   - Region 6 lies above both curves.

5. **The x-axis and y-axis**: These axes are the standard Cartesian coordinate axes. The x-axis is labeled as \( y = 0 \) and the y-axis is labeled as \( x = 0 \).

Would you like further clarification on a specific aspect of this diagram, or would you like to calculate specific points like the intersection of the line and parabola?

Here are some questions to expand upon this topic:

1. How do you calculate the points of intersection between the parabola and the line?
2. What are the characteristics of the parabola \( y = x^2 \) in terms of symmetry, vertex, and direction of opening?
3. How can you determine the area between the parabola and the line?
4. What is the geometric significance of the regions numbered 1 through 7?
5. How would the graph change if the equation of the line were altered to \( y = 2x + 4 \)?

**Tip:** To solve for the intersection points between a line and a parabola, set their equations equal to each other and solve the resulting quadratic equation.

y = x^2 with a line y = x + 4 and numbered regions

Explore the graph of the quadratic function y = x^2 and the linear equation y = x + 4, including the points of intersection and the regions divided by these curves. Learn how to solve the system of equations and analyze the graph. Suitable for Grades 9-11 students.

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