To graph the parabola using the given information:

Given:

• Vertex: (0, 3)
• Point on the parabola: $(-2, -8)$

Step-by-Step Process:

1. Identify the form of the parabola's equation: Since you are given the vertex, the vertex form of a quadratic equation is: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex of the parabola.

2. Substitute the vertex coordinates: Given the vertex is (0, 3), substitute $h = 0$ and $k = 3$: $y = a(x - 0)^2 + 3 \quad \text{or} \quad y = ax^2 + 3$

3. Use the additional point to find the value of $a$: Substitute the coordinates of the point $(-2, -8)$ into the equation: $-8 = a(-2)^2 + 3$ Simplify the equation to solve for $a$: $-8 = 4a + 3$ Subtract 3 from both sides: $-11 = 4a$ Divide by 4: $a = -\frac{11}{4}$

4. Write the final equation: The equation of the parabola is: $y = -\frac{11}{4}x^2 + 3$

5. Graph the parabola:

   • Plot the vertex at (0, 3).
   • Plot the point $(-2, -8)$.
   • Since $a < 0$, the parabola opens downward.
   • Draw a smooth curve through the points.

You can use the graphing tool provided to plot the vertex and the point, then draw the parabola based on these points.

Would you like a more detailed explanation of any of these steps?

Here are some related questions:

1. What is the standard form of a quadratic equation?
2. How does the sign of the coefficient $a$ affect the direction of the parabola?
3. What are other forms of the quadratic equation besides vertex form?
4. How do you find the axis of symmetry in a parabola?
5. Can you determine the roots of the parabola from the vertex form?

Tip: Always verify the direction of your parabola by checking the sign of the leading coefficient $a$. If $a$ is negative, the parabola opens downwards; if positive, it opens upwards.