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We are asked to find the vertex of the quadratic function \( f(x) = -5x^2 + 11x + 6 \) using the vertex formula.

The vertex formula for a quadratic function \( f(x) = ax^2 + bx + c \) is given by:

\[
x = \frac{-b}{2a}
\]

In this case:
- \( a = -5 \)
- \( b = 11 \)
- \( c = 6 \) (though we don't need \( c \) for finding the vertex's x-coordinate)

### Step 1: Calculate the x-coordinate of the vertex

\[
x = \frac{-11}{2(-5)} = \frac{-11}{-10} = \frac{11}{10}
\]

### Step 2: Calculate the y-coordinate

To find the y-coordinate, substitute \( x = \frac{11}{10} \) back into the quadratic function \( f(x) \):

\[
f\left(\frac{11}{10}\right) = -5\left(\frac{11}{10}\right)^2 + 11\left(\frac{11}{10}\right) + 6
\]

\[
f\left(\frac{11}{10}\right) = -5 \times \frac{121}{100} + 11 \times \frac{11}{10} + 6
\]

Simplifying each term:

\[
f\left(\frac{11}{10}\right) = -\frac{605}{100} + \frac{121}{10} + 6
\]

Convert all terms to have the same denominator (100):

\[
f\left(\frac{11}{10}\right) = -\frac{605}{100} + \frac{1210}{100} + \frac{600}{100}
\]

Now, sum them:

\[
f\left(\frac{11}{10}\right) = \frac{-605 + 1210 + 600}{100} = \frac{1205}{100} = 12.05
\]

### Conclusion

The vertex of the quadratic function \( f(x) = -5x^2 + 11x + 6 \) is \( \left( \frac{11}{10}, 12.05 \right) \).

Would you like further details or explanations on this? Here are 5 related questions for further exploration:

1. How do you interpret the vertex of a quadratic function geometrically?
2. What is the significance of the sign of the \( a \)-coefficient in determining the shape of the parabola?
3. How do you determine if the vertex is a maximum or minimum point?
4. Can you find the axis of symmetry for this quadratic function?
5. How would you solve for the roots of this quadratic function?

**Tip**: The x-coordinate of the vertex always lies on the axis of symmetry of the parabola.

Use the quadratic function f(x) = -5x^2 + 11x + 6 to answer the following questions. Use the vertex formula to determine the vertex.

Learn how to find the vertex of the quadratic function f(x) = -5x^2 + 11x + 6 using the vertex formula. This problem covers key algebraic concepts, including how to apply the formula x = -b / 2a. Suitable for students in Grades 9-11.

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