Let's start by analyzing the equation of the graph:

Given equation: $y = x^3 - 3x + 1$

Step 1: Find the intercepts

a) Y-intercept:

To find the y-intercept, set $x = 0$:

$y = (0)^3 - 3(0) + 1 = 1$

So, the y-intercept is at $(0, 1)$.

b) X-intercepts:

To find the x-intercepts, set $y = 0$ and solve for $x$:

$0 = x^3 - 3x + 1$

This cubic equation doesn't factor easily, so we may need to use numerical or graphical methods to find the roots. However, for simplicity, let's look at the general behavior of the function. The roots are approximately:

$x \approx -1.879, \quad x \approx 0.532, \quad x \approx 1.347$

So, the x-intercepts are approximately $(-1.879, 0)$, $(0.532, 0)$, and $(1.347, 0)$.

Step 2: Find the turning points

To find the turning points (local maxima and minima), we need to take the derivative of the function:

$y' = \frac{d}{dx}(x^3 - 3x + 1) = 3x^2 - 3$

Now, set the derivative equal to zero to find the critical points:

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

We have two critical points: $x = 1$ and $x = -1$.

To determine whether these are maxima or minima, we take the second derivative of the function:

$y'' = \frac{d}{dx}(3x^2 - 3) = 6x$

• At $x = 1$, $y''(1) = 6(1) = 6$, which is positive, indicating a local minimum at $(1, y(1))$.
• At $x = -1$, $y''(-1) = 6(-1) = -6$, which is negative, indicating a local maximum at $(-1, y(-1))$.

Now, calculate the y-values at these points:

• At $x = 1$, $y = 1^3 - 3(1) + 1 = -1$, so the local minimum is at $(1, -1)$.
• At $x = -1$, $y = (-1)^3 - 3(-1) + 1 = 3$, so the local maximum is at $(-1, 3)$.

Step 3: Summary of key points

• Y-intercept: $(0, 1)$
• X-intercepts: $(-1.879, 0)$, $(0.532, 0)$, and $(1.347, 0)$
• Turning points:
  • Local maximum at $(-1, 3)$
  • Local minimum at $(1, -1)$

Graph Description

• The graph will be a cubic curve with one local maximum at $(-1, 3)$ and one local minimum at $(1, -1)$.
• It will cross the x-axis at approximately $(-1.879, 0)$, $(0.532, 0)$, and $(1.347, 0)$, and it will pass through the y-axis at $(0, 1)$.

If you'd like, I can generate the graph for you. Would you like me to proceed with that? Or do you need more clarification on any of the steps?