You provided two equations:

1. \( y = x^2 \)
2. \( x = y^2 \)

These represent two curves. Let’s explore their relationship:

### 1. \( y = x^2 \) (a parabola):
- This is a parabola that opens upwards.
- For every value of \( x \), there is one corresponding value of \( y \).

### 2. \( x = y^2 \) (a sideways parabola):
- This is a parabola that opens to the right.
- For every value of \( y \), there is one corresponding value of \( x \).

### Intersection Points
To find where these curves intersect, we can substitute \( y = x^2 \) into \( x = y^2 \):

\[
x = (x^2)^2 = x^4
\]
This simplifies to:

\[
x = x^4
\]

Rearranging the equation:

\[
x^4 - x = 0
\]
\[
x(x^3 - 1) = 0
\]
This gives two solutions:
- \( x = 0 \)
- \( x^3 = 1 \Rightarrow x = 1 \)

### Corresponding \( y \)-values:
For \( x = 0 \), substitute into \( y = x^2 \):

\[
y = 0^2 = 0
\]

For \( x = 1 \), substitute into \( y = x^2 \):

\[
y = 1^2 = 1
\]

### Intersection Points:
The curves intersect at two points:
- \( (0, 0) \)
- \( (1, 1) \)

### Summary:
The curves \( y = x^2 \) and \( x = y^2 \) intersect at two points: \( (0, 0) \) and \( (1, 1) \).

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Would you like more details on this? Let me know if you have any questions.

Here are 5 related questions:
1. What would happen if the parabola equations were shifted?
2. Can you find the area enclosed by these curves between their intersection points?
3. What is the geometric significance of the intersection points?
4. How do these curves behave as \( x \to \infty \)?
5. Can we extend this analysis to three dimensions?

**Tip:** To solve intersection problems, substitution is a useful technique, especially when dealing with two equations!

Explore how to solve the system of equations y = x² and x = y², which represents a parabola and a sideways parabola. Learn to find their intersection points using substitution and solve polynomial equations. Suitable for high school students in Grades 9-12.

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