The problem requires finding the equation of the parabola shown in the graph, with a given format of $y = a(x - p)(x - q)$, where $a$, $p$, and $q$ are constants.

Steps to Determine the Equation

1. Identify the Roots (x-intercepts): The x-intercepts are points where the parabola crosses the x-axis, specifically:

   • $(-1, 0)$ and $(2, 0)$.

   This gives us the values of $p$ and $q$:

   • $p = -1$
   • $q = 2$

   So, the equation becomes: $y = a(x + 1)(x - 2)$

2. Determine the Vertex to Find $a$: The vertex is given at $(1, 6)$. We can use this point to find the value of $a$.

   Substitute $x = 1$ and $y = 6$ into the equation: $6 = a(1 + 1)(1 - 2)$ Simplify the equation: $6 = a(2)(-1)$ $6 = -2a$ Solving for $a$: $a = -3$

3. Write the Final Equation: Substitute $a = -3$ back into the equation: $y = -3(x + 1)(x - 2)$

Final Answer

The equation of the parabola is: $y = -3(x + 1)(x - 2)$

Would you like more details on any step, or have any questions?

Here are five related questions you might find interesting:

1. How do you find the vertex of a parabola given its equation in standard form?
2. What is the significance of the coefficient $a$ in determining the direction of a parabola?
3. How can you convert a parabola's equation from factored form to standard form?
4. How does changing the values of $p$ and $q$ affect the graph of a parabola?
5. What are some real-world applications of parabolic equations?

Tip: When working with parabolas, identifying the vertex and intercepts can make it easier to write the equation in different forms.