The equation you uploaded is:

$y = x^3 - 3x + 3$

This is a cubic function, and its general shape is determined by the cubic term $x^3$. Let's break down what we can do with this function:

1. Derivative for Critical Points: To find the critical points (where the slope is zero, i.e., possible maxima, minima, or inflection points), we can compute the derivative $\frac{dy}{dx}$.

   $\frac{dy}{dx} = 3x^2 - 3$

   Setting this equal to zero to find critical points:

   $3x^2 - 3 = 0$ $x^2 = 1$ $x = \pm 1$

   So, the critical points occur at $x = 1$ and $x = -1$.

2. Second Derivative for Concavity: The second derivative helps us determine the concavity and whether these critical points are maxima or minima.

   $\frac{d^2y}{dx^2} = 6x$

   • At $x = 1$, $\frac{d^2y}{dx^2} = 6 \times 1 = 6$ (positive, so it's a local minimum).
   • At $x = -1$, $\frac{d^2y}{dx^2} = 6 \times (-1) = -6$ (negative, so it's a local maximum).

3. Function Behavior: At large positive or negative values of $x$, the $x^3$ term dominates, so the function tends towards positive infinity as $x \to \infty$ and negative infinity as $x \to -\infty$.

4. Y-intercept: The y-intercept occurs where $x = 0$. Substituting into the equation:

   $y = 0^3 - 3(0) + 3 = 3$

   So, the y-intercept is at $(0, 3)$.

Would you like further details or clarification on any part of this solution?

Here are five related questions for further exploration:

1. What is the nature of the inflection point in this cubic function?
2. How would the graph of this function change if we added another term like $x^2$?
3. Can you find the exact coordinates of the local maximum and minimum points?
4. How would you calculate the area under the curve between two points using integration?
5. What are the asymptotic behaviors of this cubic function?

Tip: When studying functions, analyzing the first and second derivatives can give you valuable insights into their behavior and shape.