The given function is:

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

This is a quadratic function in the form \( f(x) = ax^2 + bx + c \), where:
- \(a = 2\)
- \(b = -11\)
- \(c = 2\)

### Key Features of the Function:

1. **Direction of the parabola**: Since the coefficient of \(x^2\) (which is \(a = 2\)) is positive, the parabola opens upwards.
2. **Vertex**: The vertex of a parabola is given by the formula:

\[
x_{\text{vertex}} = \frac{-b}{2a} = \frac{-(-11)}{2(2)} = \frac{11}{4} = 2.75
\]

To find the \(y\)-coordinate of the vertex, substitute \(x = 2.75\) into the original function:

\[
f(2.75) = 2 - 11(2.75) + 2(2.75)^2 = 2 - 30.25 + 15.125 = -13.125
\]

So, the vertex is at \( (2.75, -13.125) \).

3. **Axis of Symmetry**: The axis of symmetry is the vertical line through the vertex, \(x = 2.75\).

4. **Y-intercept**: This occurs when \(x = 0\):

\[
f(0) = 2 - 11(0) + 2(0)^2 = 2
\]

Thus, the y-intercept is \( (0, 2) \).

5. **Roots**: To find the x-intercepts (roots), set \(f(x) = 0\):

\[
2x^2 - 11x + 2 = 0
\]

This is a quadratic equation. You can solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values \(a = 2\), \(b = -11\), and \(c = 2\):

\[
x = \frac{11 \pm \sqrt{(-11)^2 - 4(2)(2)}}{2(2)} = \frac{11 \pm \sqrt{121 - 16}}{4} = \frac{11 \pm \sqrt{105}}{4}
\]

The exact solutions are:

\[
x = \frac{11 \pm \sqrt{105}}{4}
\]

These are the roots of the equation.

Would you like more details or have any questions?

Here are some related questions you can try:

1. How do you complete the square for this quadratic function?
2. What is the minimum value of \(f(x)\), and at what point does it occur?
3. Can you factor the quadratic expression \(2x^2 - 11x + 2\)?
4. What is the domain and range of this function?
5. How would you graph this function step by step?

**Tip**: The discriminant \((b^2 - 4ac)\) tells you about the nature of the roots. If it's positive, the quadratic has two real roots.

Learn how to solve the quadratic function f(x) = 2 - 11x + 2x² by finding the vertex, axis of symmetry, y-intercept, and roots using the quadratic formula. Explore key algebraic concepts for Grades 9-10.

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