Consider the given function \( d(x)=3 x^{2}-12 x+11 \). (a) Write the function in vertex form. (b) Identify the vertex. (c) Determine the \( x \)-intercept(s). (d) Determine the \( y \)-intercept(s). (e) Sketch the function. (f) Determine the axis of symmetry. (g) Determine the minimum or maximum value of the function. (h) Write the domain and range in interval notation. Write your answers in exact form.

To tackle the given function \( d(x)=3 x^{2}-12 x+11 \): (a) To convert to vertex form, we complete the square. \[ d(x) = 3(x^2 - 4x) + 11 \] Complete the square inside the parentheses: \[ d(x) = 3 \left( (x - 2)^2 - 4 \right) + 11 \] This simplifies to: \[ d(x) = 3(x - 2)^2 - 12 + 11 \] \[ d(x) = 3(x - 2)^2 - 1 \] (b) The vertex form reveals that the vertex is at \( (2, -1) \). (c) To find the \( x \)-intercept(s), set \( d(x) = 0 \): \[ 3x^2 - 12x + 11 = 0 \] Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 3 \cdot 11}}{2 \cdot 3} \] Calculating the discriminant: \[ 144 - 132 = 12 \] Thus, \[ x = \frac{12 \pm \sqrt{12}}{6} = \frac{12 \pm 2\sqrt{3}}{6} = 2 \pm \frac{\sqrt{3}}{3} \] The \( x \)-intercepts are \( \left( 2 + \frac{\sqrt{3}}{3}, 0 \right) \) and \( \left( 2 - \frac{\sqrt{3}}{3}, 0 \right) \). (d) To find the \( y \)-intercept, set \( x = 0 \): \[ d(0) = 3(0)^2 - 12(0) + 11 = 11 \] So, the \( y \)-intercept is \( (0, 11) \). (e) A sketch would show a parabola opening upwards, with the vertex at \( (2, -1) \), crossing the \( y \)-axis at \( (0, 11) \), and the \( x \)-intercepts calculated earlier. (f) The axis of symmetry is given by \( x = h \) where \( h = 2 \). (g) The function has a minimum value at the vertex, so the minimum value is \( -1 \). (h) The domain of a quadratic function is all real numbers: \[ (-\infty, \infty) \] The range starts from the minimum value to infinity: \[ [-1, \infty) \] So, the complete solution is: - Vertex form: \( d(x) = 3(x - 2)^2 - 1 \) - Vertex: \( (2, -1) \) - \( x \)-intercepts: \( \left( 2 + \frac{\sqrt{3}}{3}, 0 \right), \left( 2 - \frac{\sqrt{3}}{3}, 0 \right) \) - \( y \)-intercept: \( (0, 11) \) - Axis of symmetry: \( x = 2 \) - Minimum value: \( -1 \) - Domain: \( (-\infty, \infty) \) - Range: \( [-1, \infty) \)