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To write the equation of the parabola in standard form based on the graph, we need to recognize the general characteristics of the parabola.

### Observations from the graph:
- The parabola is **opening upwards**.
- The **vertex** of the parabola is at \( (0, 4) \).
- The parabola passes through the points \( (-4, 0) \), \( (-0.5, 0) \), and \( (0, 4) \), which gives us some key points for determining the equation.

### Standard form of a parabola:
The standard form of a parabola's equation is given by:
\[
y = a(x - h)^2 + k
\]
where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) is a constant that determines the width and direction of the parabola.

### Step 1: Substituting the vertex
From the graph, the vertex is \( (0, 4) \). So, the equation becomes:
\[
y = a(x - 0)^2 + 4 \quad \Rightarrow \quad y = a x^2 + 4
\]

### Step 2: Using a point on the graph
We can use one of the points that the parabola passes through to solve for \( a \). Let's use \( (-4, 0) \), a point on the parabola.

Substitute \( x = -4 \) and \( y = 0 \) into the equation:
\[
0 = a(-4)^2 + 4
\]
\[
0 = 16a + 4
\]
\[
16a = -4
\]
\[
a = -\frac{1}{4}
\]

### Final Equation:
Thus, the equation of the parabola is:
\[
y = -\frac{1}{4}x^2 + 4
\]

Would you like more details on how this equation was derived or further explanation on any steps?

Here are 5 related questions for practice:
1. How do you convert the equation to vertex form if the vertex is different?
2. How do you find the focus of a parabola from the standard form?
3. What is the axis of symmetry for this parabola?
4. How would the equation change if the parabola opened downwards?
5. How can you determine the direction of a parabola from its equation?

**Tip**: The value of \( a \) in the quadratic equation determines how "wide" or "narrow" the parabola is—smaller values of \( a \) make it wider, while larger values make it narrower.

Write a standard form function based on the given graph of the parabola.

Learn how to derive the standard form of a quadratic equation based on the graph of a parabola. This problem includes using the vertex form, substituting key points like (-4, 0), and finding the coefficient 'a'. Suitable for students in Grades 9-12.

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