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the vertex is (0,3) give me the vertex equation of this graph show all work

Given that the vertex of the parabola is at \((0, 3)\) and the graph represents a downward-opening parabola (based on its curvature), the vertex form of the equation of a parabola is:

\[
y = a(x - h)^2 + k
\]

where:
- \((h, k)\) is the vertex of the parabola,
- \(a\) controls the direction and the "width" of the parabola (whether it is wide or narrow).

Here, \((h, k) = (0, 3)\), so we can substitute these values into the vertex form:

\[
y = a(x - 0)^2 + 3
\]
\[
y = a x^2 + 3
\]

Next, we need to determine the value of \(a\). From the graph, the parabola passes through the point \((1, 0)\), as it appears the graph touches the x-axis at this point. Substituting this point into the equation will allow us to solve for \(a\):

\[
0 = a(1)^2 + 3
\]
\[
0 = a + 3
\]
\[
a = -3
\]

Thus, the equation of the parabola is:

\[
y = -3x^2 + 3
\]

### Final Answer:
The vertex form of the equation is:
\[
y = -3x^2 + 3
\]

Would you like further explanation or have any questions? Here are five questions to extend your understanding:

1. How does the value of \(a\) affect the width and direction of the parabola?
2. What changes in the equation if the vertex is shifted from \((0, 3)\) to a different point?
3. Can you identify the axis of symmetry from this equation?
4. How do you find the x-intercepts of this parabola?
5. How would the graph change if \(a\) were positive?

**Tip:** Always substitute known points into the equation to solve for constants like \(a\) when you only know partial information about a graph!

Learn how to find the vertex equation of a parabola with a vertex at (0, 3). This problem covers key algebraic concepts including vertex form of quadratic equations and determining the coefficient 'a'. Suitable for students in Grades 9-11.

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