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The graph shows a quadratic function with intercepts at \( (-2, 0) \) and \( (6, 0) \), and the vertex is at \( (4, -3) \). To write the quadratic function in standard form, \( f(x) = ax^2 + bx + c \), we can follow these steps:

### Step 1: Identify the factored form of the quadratic.
Since the x-intercepts are \( x = -2 \) and \( x = 6 \), the function in factored form is:
\[
f(x) = a(x + 2)(x - 6)
\]
where \( a \) is a constant to be determined.

### Step 2: Use the vertex to find \( a \).
We know the vertex is \( (4, -3) \), so substitute \( x = 4 \) and \( f(x) = -3 \) into the equation to solve for \( a \):
\[
-3 = a(4 + 2)(4 - 6)
\]
\[
-3 = a(6)(-2)
\]
\[
-3 = -12a
\]
\[
a = \frac{-3}{-12} = \frac{1}{4}
\]

### Step 3: Write the equation in factored form.
Now that we know \( a = \frac{1}{4} \), the function in factored form is:
\[
f(x) = \frac{1}{4}(x + 2)(x - 6)
\]

### Step 4: Expand to write in standard form.
To convert this into standard form, expand the factors:
\[
f(x) = \frac{1}{4}(x^2 - 6x + 2x - 12)
\]
\[
f(x) = \frac{1}{4}(x^2 - 4x - 12)
\]
\[
f(x) = \frac{1}{4}x^2 - \frac{1}{4}(4x) - \frac{1}{4}(12)
\]
\[
f(x) = \frac{1}{4}x^2 - x - 3
\]

Thus, the quadratic function in standard form is:
\[
f(x) = \frac{1}{4}x^2 - x - 3
\]

Would you like more details or have any further questions?

Here are 5 related questions to explore further:
1. How do you convert a quadratic function from vertex form to standard form?
2. What is the significance of the vertex in a quadratic function?
3. How do you determine the direction in which a parabola opens?
4. How can you find the axis of symmetry of a parabola?
5. What methods can be used to find the roots of a quadratic equation?

**Tip**: Remember, the vertex form of a quadratic equation is particularly useful when you know the vertex, making it easier to graph.

Write a quadratic function in standard form whose graph is shown with intercepts at (-2, 0) and (6, 0), and a vertex at (4, -3).

Learn how to write a quadratic function in standard form from a graph with intercepts at (-2, 0) and (6, 0) and vertex at (4, -3). Explore the steps to determine the factored form, solve for 'a,' and convert the function to standard form.

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