To solve this problem, we need to determine the points of intersection between the two equations in the system:

1. $y = 2x$ (a linear equation representing a line)
2. $y = x^2 + 1$ (a quadratic equation representing a parabola)

Step 1: Set the Equations Equal to Each Other

To find where these graphs intersect, set the right-hand sides of the equations equal to each other:

$2x = x^2 + 1$

Step 2: Rearrange to Form a Quadratic Equation

Rewrite this equation in standard quadratic form:

$x^2 - 2x + 1 = 0$

Step 3: Simplify and Solve

The equation can be factored as follows:

$(x - 1)^2 = 0$

So, we have:

$x = 1$

Step 4: Substitute $x = 1$ Back to Find $y$

Using $x = 1$ in either of the original equations (we'll use $y = 2x$):

$y = 2 \cdot 1 = 2$

Thus, the intersection point is $(1, 2)$.

Step 5: Analyze the Graphs

1. The line $y = 2x$ will be a straight line passing through the origin with a slope of 2.
2. The parabola $y = x^2 + 1$ opens upwards and has its vertex at $(0, 1)$.

The graphs intersect at the point $(1, 2)$.

Conclusion

Look for the graph that has:

• A straight line with a positive slope (rising from left to right).
• A parabola that opens upward with its vertex at $(0, 1)$.
• The intersection point at $(1, 2)$.

Would you like further details on plotting or interpreting this system?

Here are some questions to explore further:

1. How do you determine the points of intersection for other types of curves?
2. What happens if the line had a different slope?
3. How would the solution change if the parabola opened downward?
4. What if we had two quadratic equations? How would we find intersections then?
5. How can this technique be extended to systems with more than two equations?

Tip: When analyzing systems graphically, finding intersections algebraically first can help you identify important points on the graph!