\#4 \( f(x)=x^{2}-12 x+11 \)

**Step 1.** Write the quadratic function: \[ f(x)=x^{2}-12x+11 \] **Step 2.** Factor the quadratic expression. We look for two numbers whose product is \(11\) and whose sum is \(-12\). These numbers are \(-1\) and \(-11\) because: \[ (-1)\times(-11)=11 \quad \text{and} \quad (-1)+(-11)=-12. \] Thus, the factored form is: \[ f(x) = (x-1)(x-11). \] **Step 3.** Find the zeros of the function. Set \( f(x)=0 \): \[ (x-1)(x-11)=0. \] Thus, the solutions are: \[ x-1=0 \quad \Rightarrow \quad x=1, \] \[ x-11=0 \quad \Rightarrow \quad x=11. \] **Step 4.** Convert the function into vertex form by completing the square. Start with: \[ f(x)=x^{2}-12x+11. \] Group the quadratic and linear terms and complete the square: \[ x^{2}-12x = (x-6)^{2}-36. \] Now, substitute back into the function: \[ f(x) = (x-6)^{2} - 36 + 11 = (x-6)^{2} - 25. \] Thus, the vertex form is: \[ f(x)=(x-6)^{2}-25. \] **Step 5.** Identify the vertex and other characteristics. In the form: \[ f(x)=(x-6)^{2}-25, \] the vertex is at: \[ (6,-25). \] Since the coefficient of \((x-6)^{2}\) is positive, the parabola opens upward, and the vertex represents the minimum point of the function. The axis of symmetry is the vertical line: \[ x=6. \] **Summary of the results:** - **Factored Form:** \((x-1)(x-11)\) - **Zeros:** \(x=1\) and \(x=11\) - **Vertex Form:** \((x-6)^{2}-25\) - **Vertex:** \((6,-25)\) - **Axis of Symmetry:** \(x=6\)