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To write a quadratic function \( f(x) \) whose only zero is \(-11\), we need to create a function that has this as a repeated root. Since a quadratic function can be factored as \( f(x) = a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots, having a single root of \(-11\) implies that both \( r_1 \) and \( r_2 \) are \(-11\).

Thus, the function can be written as:

\[
f(x) = a(x + 11)(x + 11)
\]

Simplifying, we get:

\[
f(x) = a(x + 11)^2
\]

For simplicity, let's set \( a = 1 \):

\[
f(x) = (x + 11)^2
\]

Expanding this, we have:

\[
f(x) = x^2 + 22x + 121
\]

Therefore, the quadratic function with only \(-11\) as its zero is:

\[
f(x) = x^2 + 22x + 121
\]

Would you like more details, or do you have any questions?

Here are some related questions you might find helpful:

1. What is the importance of the leading coefficient \( a \) in a quadratic function?
2. How do you determine the vertex of the quadratic function \( f(x) = (x + 11)^2 \)?
3. How would the graph of \( f(x) = x^2 + 22x + 121 \) look like?
4. What are the implications of having a double root in a quadratic function?
5. How do you find the axis of symmetry for a quadratic function?

**Tip:** When a quadratic function has a repeated root, it touches the x-axis at that root but does not cross it.

Write a quadratic function f whose only zero is -11.

Discover how to write a quadratic function whose only zero is -11. This example illustrates key algebraic concepts, including quadratic functions, their roots, and the significance of a repeated root. Suitable for students in Grades 9-10.

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