https://file.aiquickdraw.com/mathbot/file/1728166116320ebwlj8bd.jpg

The graph provided represents a quadratic function, and we can derive its equation from key points. From the graph, the roots (x-intercepts) are \( x = 2 \) and \( x = 10 \), and the vertex is at \( (6, -4) \).

The standard form of a quadratic function is:

\[
y = ax^2 + bx + c
\]

However, we will use the factored form first, based on the roots:

\[
y = a(x - 2)(x - 10)
\]

Now, we need to determine the value of \( a \). Since the vertex is \( (6, -4) \), we can substitute \( x = 6 \) and \( y = -4 \) into the equation:

\[
-4 = a(6 - 2)(6 - 10)
\]

Simplifying:

\[
-4 = a(4)(-4)
\]

\[
-4 = -16a
\]

Solving for \( a \):

\[
a = \frac{-4}{-16} = \frac{1}{4}
\]

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

\[
y = \frac{1}{4}(x - 2)(x - 10)
\]

Now, expand this to write it in standard form:

\[
y = \frac{1}{4}[(x - 2)(x - 10)] = \frac{1}{4}(x^2 - 12x + 20)
\]

Distribute the \( \frac{1}{4} \):

\[
y = \frac{1}{4}x^2 - 3x + 5
\]

Therefore, the equation of the quadratic function in standard form is:

\[
y = \frac{1}{4}x^2 - 3x + 5
\]

Would you like further details or have any questions? Here are five related questions you might explore:

1. How can you find the vertex of a quadratic equation from its standard form?
2. What is the significance of the value of \( a \) in the quadratic equation?
3. How would the graph change if \( a \) was negative?
4. How do you convert a quadratic equation from factored form to standard form?
5. What methods can you use to find the roots of a quadratic equation?

**Tip:** Remember that the sign of \( a \) determines the direction of the parabola—positive opens upwards, negative opens downwards!

The graph represents a quadratic function. Write an equation of the function in standard form.

Learn how to write the equation of a quadratic function from its graph with roots at (2,0) and (10,0), and a vertex at (6, -4). This solution covers factored and standard forms, suitable for students in Grades 9-11.

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