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The graph you provided shows a parabolic function \( f(x) \), and we want to express this in standard form. The parabola has the following important points:

- The vertex is at \( (1, 4) \).
- Another point on the parabola is \( (3, -4) \).

### General Standard Form of a Quadratic Equation:
The standard form of a quadratic function is:
\[
f(x) = a(x - h)^2 + k
\]
Where \( (h, k) \) is the vertex, and \( a \) is a constant that determines the direction and width of the parabola.

### Step 1: Substitute the vertex \((h, k)\)
Given that the vertex is \( (1, 4) \), we have:
\[
f(x) = a(x - 1)^2 + 4
\]

### Step 2: Use the point \( (3, -4) \) to find \( a \)
Substitute \( x = 3 \) and \( f(3) = -4 \) into the equation to solve for \( a \):
\[
-4 = a(3 - 1)^2 + 4
\]
\[
-4 = a(2)^2 + 4
\]
\[
-4 = 4a + 4
\]
\[
-4 - 4 = 4a
\]
\[
-8 = 4a
\]
\[
a = -2
\]

### Step 3: Write the equation
Now that we know \( a = -2 \), the equation of the parabola in vertex form is:
\[
f(x) = -2(x - 1)^2 + 4
\]

### Step 4: Expand to Standard Form
To convert this into standard form \( f(x) = ax^2 + bx + c \), expand the equation:
\[
f(x) = -2(x^2 - 2x + 1) + 4
\]
\[
f(x) = -2x^2 + 4x - 2 + 4
\]
\[
f(x) = -2x^2 + 4x + 2
\]

Thus, the standard form of the quadratic equation is:
\[
f(x) = -2x^2 + 4x + 2
\]

Would you like more details on any part of this process?

### Related Questions:
1. How can you find the axis of symmetry from a quadratic function in standard form?
2. How would you determine the roots of this quadratic function?
3. Can you describe how the value of \( a \) affects the shape of the parabola?
4. How would the graph change if the vertex were shifted?
5. How can you convert a standard form quadratic function back into vertex form?

### Tip:
When finding the equation of a parabola, identifying the vertex and one additional point can help you quickly determine the equation.

Learn how to convert a quadratic equation from vertex form to standard form using a given vertex (1, 4) and a point (3, -4). This tutorial covers key algebraic concepts like vertex form and how to expand it to standard form. Suitable for students in Grades 9-12.

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