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The graph represents a quadratic function, and we are asked to write the equation of this function in standard form. The standard form of a quadratic equation is:

\[
y = ax^2 + bx + c
\]

### Key observations from the graph:
- The vertex of the parabola is at \( (2, -6) \).
- Another point on the parabola is \( (5, 3) \).

Since we know the vertex, we can start by using the **vertex form** of a quadratic equation, which is:

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

where \( (h, k) \) is the vertex of the parabola. From the graph, we see that \( h = 2 \) and \( k = -6 \), so the equation becomes:

\[
y = a(x - 2)^2 - 6
\]

Now, we need to determine the value of \( a \). To do this, we can substitute the point \( (5, 3) \), which lies on the parabola, into the equation and solve for \( a \).

Substitute \( x = 5 \) and \( y = 3 \):

\[
3 = a(5 - 2)^2 - 6
\]

Simplifying the equation:

\[
3 = a(3)^2 - 6
\]
\[
3 = 9a - 6
\]
\[
9 = 9a
\]
\[
a = 1
\]

Thus, the equation in vertex form is:

\[
y = (x - 2)^2 - 6
\]

### Converting to standard form:
Now, expand the equation:

\[
y = (x - 2)^2 - 6
\]
\[
y = (x^2 - 4x + 4) - 6
\]
\[
y = x^2 - 4x - 2
\]

So, the equation of the quadratic function in standard form is:

\[
y = x^2 - 4x - 2
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How do you convert a quadratic equation from vertex form to standard form?
2. What does the vertex of a parabola represent in terms of a quadratic function?
3. How do you find the axis of symmetry for a parabola from its equation?
4. Can you determine whether a quadratic function opens upwards or downwards just by looking at its equation?
5. How can you find the y-intercept of a quadratic function in standard form?

**Tip:** When given a vertex and a point, using the vertex form of the quadratic equation helps simplify finding the equation!

The graph represents a quadratic function. Write an equation of the function in standard form.

Learn how to derive the quadratic equation from a graph, starting with the vertex form and converting it into standard form. This problem uses key algebraic concepts such as vertex and standard forms of quadratic equations.

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