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To write the equation of the quadratic function in standard form based on the graph, we first analyze the key features of the graph:

### Key observations:
1. **Zeros (x-intercepts)**:
   - The graph passes through the points \((-4, 0)\), \((-0.5, 0)\), and \((0, 4)\).

2. **Vertex**:
   - The vertex of the parabola appears to be around \((-2.25, -6)\), as the graph reaches its minimum there. The graph opens upwards.

### Step-by-step solution:
The standard form of a quadratic function is:

\[
f(x) = ax^2 + bx + c
\]

Alternatively, we can use the factored form:

\[
f(x) = a(x - x_1)(x - x_2)
\]

Where \(x_1 = -4\) and \(x_2 = -0.5\) are the x-intercepts. So the factored form becomes:

\[
f(x) = a(x + 4)(x + 0.5)
\]

Now, we use the given point \((0, 4)\) to find the value of \(a\). Substituting \(x = 0\) and \(f(x) = 4\):

\[
4 = a(0 + 4)(0 + 0.5)
\]
\[
4 = a(4)(0.5)
\]
\[
4 = 2a
\]
\[
a = 2
\]

Thus, the equation of the quadratic function in factored form is:

\[
f(x) = 2(x + 4)(x + 0.5)
\]

Now, expanding this into standard form:

\[
f(x) = 2(x + 4)(x + 0.5)
\]
\[
= 2(x^2 + 0.5x + 4x + 2)
\]
\[
= 2(x^2 + 4.5x + 2)
\]
\[
= 2x^2 + 9x + 4
\]

### Final answer:
The equation of the quadratic function in standard form is:

\[
f(x) = 2x^2 + 9x + 4
\]

Would you like more details or have any questions about this solution?

### Related questions:
1. How do you determine the vertex of a parabola from the equation in standard form?
2. What is the significance of the coefficient "a" in the quadratic equation?
3. How can you convert a quadratic function from vertex form to standard form?
4. How can you identify the axis of symmetry from the quadratic equation?
5. What other methods are there to find the equation of a quadratic function from a graph?

### Tip:
When working with quadratic functions, the vertex form \(f(x) = a(x - h)^2 + k\) is particularly useful for identifying the vertex directly.

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 with zeros at (-4, 0) and (-0.5, 0) and a point at (0, 4). This problem demonstrates how to use the factored form of a quadratic equation and expand it into standard form, suitable for Grades 9-12.

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