You're given the system of equations:

1. $y^2 = 2x$
2. $x = 2y$

And you want to explore this system about the y-axis. It seems like you're interested in understanding the behavior of the graph and the related concepts like symmetry or finding the area of the region enclosed by these curves.

Step 1: Solve the system

First, let's substitute $x = 2y$ from the second equation into the first equation:

$y^2 = 2x \quad \text{becomes} \quad y^2 = 2(2y)$ $y^2 = 4y$ Now, rearrange this: $y^2 - 4y = 0$ Factor the equation: $y(y - 4) = 0$ So, $y = 0$ or $y = 4$.

Step 2: Find corresponding x-values

Now substitute $y = 0$ and $y = 4$ back into the equation $x = 2y$ to find the corresponding x-values:

• When $y = 0$, $x = 2(0) = 0$.
• When $y = 4$, $x = 2(4) = 8$.

Thus, the system intersects at the points $(0, 0)$ and $(8, 4)$.

Step 3: Analyze Symmetry

Regarding the symmetry about the y-axis, we can see that the equation $y^2 = 2x$ is symmetric with respect to the x-axis, but the line $x = 2y$ is not symmetric with respect to the y-axis. The intersection of these curves forms a small region in the first quadrant, as both x and y are positive for the given points of intersection.

Step 4: Reflection about the y-axis

To reflect the system about the y-axis, we replace $x$ with $-x$ in both equations and analyze the system:

• For $y^2 = 2x$, we get $y^2 = -2x$, which has no real solutions since $y^2$ is always non-negative.
• For $x = 2y$, replacing $x$ with $-x$, we get $-x = 2y$, which implies $x = -2y$.

Hence, the system is not symmetric about the y-axis in the conventional sense of a reflection.

Additional Thoughts:

• You might be interested in finding the area enclosed between these curves or other related quantities.
• Let me know if you want to go into more detail about how these equations behave graphically, their intersection points, or find areas.

Would you like further details on any of these steps, or perhaps something else? Here are some related questions:

1. How would we calculate the area enclosed by these curves?
2. What happens if we reflect the system about the x-axis instead of the y-axis?
3. How do these curves behave for negative values of $y$?
4. Can we generalize the process of solving systems of equations graphically?
5. How do symmetry and reflections influence the area calculation?

Tip: When analyzing symmetry, always check if substituting negative values of $x$ or $y$ maintains the structure of the equation, as this often gives quick insights into the nature of the graph.