The equation in the image provided is $y = -3x^2$, and you are asked to plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Step-by-step:

1. Vertex of the Parabola: The equation is of the form $y = ax^2$, where $a = -3$. In this case, the parabola opens downwards because $a < 0$, and the vertex is at $(0, 0)$ because there's no shift in the $x$- or $y$-directions.

2. Choosing Points: To plot five points, we need to choose two points to the left of the vertex (negative $x$-values) and two points to the right (positive $x$-values).

3. Calculating the Points: We will plug in $x = -2, -1, 0, 1, 2$ into the equation to get the corresponding $y$-values.

   • For $x = -2$: $y = -3(-2)^2 = -3(4) = -12$ Point: $(-2, -12)$

   • For $x = -1$: $y = -3(-1)^2 = -3(1) = -3$ Point: $(-1, -3)$

   • For $x = 0$: $y = -3(0)^2 = 0$ Point: $(0, 0)$ (vertex)

   • For $x = 1$: $y = -3(1)^2 = -3$ Point: $(1, -3)$

   • For $x = 2$: $y = -3(2)^2 = -12$ Point: $(2, -12)$

4. Final Points:

   • $(-2, -12)$
   • $(-1, -3)$
   • $(0, 0)$ (vertex)
   • $(1, -3)$
   • $(2, -12)$

Graphing:

You can now plot these points on a graph, which will form a symmetric, downward-opening parabola centered at $(0, 0)$.

Would you like to go into more details or have any questions?

Here are five follow-up questions:

1. How does the value of $a$ in $y = ax^2$ affect the shape of the parabola?
2. What is the effect of adding constants to $x$ or $y$ in the equation of a parabola?
3. How can you determine if a parabola opens upwards or downwards from its equation?
4. What is the significance of the vertex in a parabolic equation?
5. How can you find the axis of symmetry for a given parabola?

Tip: The vertex form of a parabola is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. This form makes it easier to identify transformations.