To sketch the curve defined by the equation $x^2y = 3$, we need to follow a few steps. Here's a step-by-step guide to help you visualize the curve:

Step 1: Solve for $y$

We can rewrite the given equation in terms of $y$: $x^2y = 3$ Dividing both sides by $x^2$, we get: $y = \frac{3}{x^2}$ This is an equation of a reciprocal function, which means it will have certain characteristics such as asymptotes and symmetry.

Step 2: Analyze the function

1. Asymptotes:

   • Since $y = \frac{3}{x^2}$, there is a vertical asymptote at $x = 0$, because the function is undefined when $x = 0$.
   • As $|x|$ increases (moves further away from 0), $y$ approaches 0, so there is a horizontal asymptote at $y = 0$.

2. Symmetry:

   • The equation $y = \frac{3}{x^2}$ is symmetric with respect to the y-axis because $x^2$ is the same for both positive and negative values of $x$. So, the graph will be the same for $x > 0$ and $x < 0$.

Step 3: Plot key points

To understand the shape better, let's find some values for $y$ at specific $x$ values:

• When $x = 1$, $y = \frac{3}{1^2} = 3$.
• When $x = 2$, $y = \frac{3}{2^2} = \frac{3}{4} = 0.75$.
• When $x = -1$, $y = \frac{3}{(-1)^2} = 3$.
• When $x = -2$, $y = \frac{3}{(-2)^2} = 0.75$.

Step 4: Shape of the curve

• The curve is hyperbolic in nature, with the following characteristics:
  • The curve will approach but never touch the x-axis (horizontal asymptote at $y = 0$).
  • The curve will have symmetry across the y-axis.
  • As $x \to \infty$ or $x \to -\infty$, $y \to 0$.

Summary of the sketch:

• The curve will have two branches, one in the first quadrant (for $x > 0$) and one in the second quadrant (for $x < 0$), both approaching the x-axis but never touching it.

Would you like a visual plot of this curve?

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5 Related Questions:

1. What are asymptotes and how are they determined for different functions?
2. How does symmetry affect the shape of a graph?
3. What happens to a function when dividing by a squared term?
4. How can we determine critical points of a function for curve sketching?
5. What are the real-world applications of reciprocal functions?

Tip:

When dealing with reciprocal functions, always check for asymptotes and symmetry to better understand the behavior of the graph!