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To determine the standard form of the quadratic function \( y = ax^2 + bx + c \) given the vertex and the values for \( a \), \( b \), and \( c \), we can use the vertex form and convert it to standard form.

### Given Data:
- **Vertex**: \( (-3, 1) \)
- **Values**: \( a = \frac{1}{3} \), \( b = 2 \), \( c = 4 \)

### Step-by-Step Solution:

1. **Vertex Form**:
   The vertex form of a quadratic equation is:
   \[
   y = a(x - h)^2 + k
   \]
   where \( (h, k) \) is the vertex. Here, \( h = -3 \) and \( k = 1 \).

   Substituting \( a = \frac{1}{3} \), \( h = -3 \), and \( k = 1 \):
   \[
   y = \frac{1}{3}(x + 3)^2 + 1
   \]

2. **Expanding to Standard Form**:
   Now, expand \( (x + 3)^2 \) and simplify to get the standard form \( y = ax^2 + bx + c \).

   \[
   y = \frac{1}{3}(x^2 + 6x + 9) + 1
   \]
   \[
   y = \frac{1}{3}x^2 + 2x + 3 + 1
   \]
   \[
   y = \frac{1}{3}x^2 + 2x + 4
   \]

3. **Conclusion**:
   The standard form of the function is:
   \[
   y = \frac{1}{3}x^2 + 2x + 4
   \]

So, the values in the question match the equation derived here. 

Would you like further explanation on the expansion or on identifying parts of the vertex form?

### Related Questions:
1. How do you convert from vertex form to standard form for other values of \( a \), \( h \), and \( k \)?
2. What does the value of \( a \) indicate about the parabola's direction and width?
3. How can you find the vertex from the standard form of a quadratic equation?
4. What role does the value of \( b \) play in determining the shape of the parabola?
5. How do you complete the square to convert from standard form to vertex form?

### Tip:
Always double-check each term when expanding squares to avoid small errors, as these can affect the entire equation.

What is the standard form of the quadratic function y = ax^2 + bx + c shown in the graph below?

Learn how to determine the standard form of a quadratic function y = ax^2 + bx + c from a graph. This problem helps understand key algebraic concepts, including identifying values of a, b, and c from a quadratic graph. Suitable for Grades 9-11 students.

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