of turning points, degree?, leading coefficient?, f(1)=, x intercepts/ zeros, y intercepts, minima, maxima,increasing and decreasing, domain, range, double roots?

Let's analyze the given graph step by step:

1. Number of Turning Points:
   The graph has 2 turning points (one maximum and one minimum).

   • A peak at around $x = -2$ and a valley at around $x = 0$.

2. Degree of the Polynomial:
   The graph shows the behavior of a cubic function (degree 3). This is because it has two turning points and ends in opposite directions (down on the left, up on the right).

3. Leading Coefficient:
   Since the left end of the graph goes to $-\infty$ and the right end goes to $+\infty$, the leading coefficient is positive.

4. $f(1)$:
   At $x = 1$, the graph appears to be around $y = 3$. So, $f(1) \approx 3$.

5. X-Intercepts (Zeros):
   The graph crosses the x-axis at $x = -3$ and $x = 0$. So, the x-intercepts are $x = -3$ and $x = 0$.

6. Y-Intercept:
   The graph crosses the y-axis at $y = 0$. So, the y-intercept is $(0, 0)$.

7. Local Minima and Maxima:

   • Local Maximum at $x \approx -2$ with $y \approx 4$.
   • Local Minimum at $x = 0$ with $y = 0$.

8. Increasing and Decreasing Intervals:

   • Increasing: $(-\infty, -2)$ and $(0, \infty)$
   • Decreasing: $(-2, 0)$

9. Domain:
   The graph extends infinitely left and right, so the domain is $(-\infty, \infty)$.

10. Range:
    The graph also extends infinitely up and down, so the range is $(-\infty, \infty)$.

11. Double Roots:
    At $x = 0$, the graph touches the x-axis but does not cross it, indicating a double root at $x = 0$.

Would you like me to explain any part of this analysis in more detail?

Related Questions:

1. How can we find the exact equation of this cubic function?
2. What is the effect of changing the leading coefficient on the graph's shape?
3. How are turning points related to the degree of a polynomial?
4. What is the relationship between multiplicity of roots and how the graph crosses or touches the x-axis?
5. How can we determine concavity and inflection points for this function?

Tip: The number of turning points of a polynomial is at most $n - 1$, where $n$ is the degree of the polynomial.