The graph provided shows a parabola, and the task is to write the quadratic function in standard form. The standard form of a quadratic function is:

$y = ax^2 + bx + c$

From the graph, we can identify key points to find the equation:

• The vertex is at $(-2, -3)$, which suggests using the vertex form of a parabola: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex. In this case, $(h, k) = (-2, -3)$.

Thus, we start with: $y = a(x + 2)^2 - 3$

• We also know the parabola passes through the point $(0, 1)$, which can be used to solve for $a$. Plugging in this point $(x = 0, y = 1)$:

$1 = a(0 + 2)^2 - 3$ $1 = a(4) - 3$ $4a = 4 \implies a = 1$

So, the equation of the parabola is: $y = (x + 2)^2 - 3$

Expanding this to standard form: $y = (x^2 + 4x + 4) - 3$ $y = x^2 + 4x + 1$

Thus, the standard form of the quadratic function is:

$y = x^2 + 4x + 1$

Would you like further details or have any questions?

Here are some related questions:

1. How do we convert a quadratic function from vertex form to standard form?
2. How can we find the vertex of a parabola if it's given in standard form?
3. What are other methods to determine the quadratic equation from a graph?
4. How does the value of $a$ affect the shape of the parabola?
5. How can we determine the axis of symmetry from the standard form of a quadratic?

Tip: The coefficient $a$ in a quadratic equation controls the "width" and direction of the parabola: if $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards.