1. **State the problem:** We are given the function $F(x,y) = 3x^2 - y^2 - 1$ and want to understand its form and behavior. 2. **Formula and explanation:** This is a quadratic function in two variables $x$ and $y$. It can be seen as a difference of squares with coefficients: $3x^2$ is positive and $-y^2$ is negative. 3. **Rewrite the function:** $$F(x,y) = 3x^2 - y^2 - 1$$ 4. **Interpretation:** This function represents a hyperbolic paraboloid surface in 3D space. 5. **Find intercepts:** - Set $F(x,y) = 0$ to find the level curve: $$3x^2 - y^2 - 1 = 0 \implies y^2 = 3x^2 - 1$$ - For real $y$, $3x^2 - 1 \geq 0 \implies |x| \geq \frac{1}{\sqrt{3}}$ 6. **Summary:** The function $F(x,y)$ is a saddle-shaped surface with a hyperbolic paraboloid form. **Final answer:** The function $F(x,y) = 3x^2 - y^2 - 1$ describes a hyperbolic paraboloid surface.