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Completing the square is a method used to convert a quadratic function into its vertex form. Starting with the quadratic function given: $$f(x) = x^2 - 8x + 12$$ We focus on the quadratic and linear terms. Separate them from the constant: $$f(x) = (x^2 - 8x) + 12$$ To complete the square, we add and then subtract the square of half the coefficient of x inside the parentheses. First, find half of the coefficient of x, which is -8: $$\frac{-8}{2} = -4$$ Next, square -4 to get 16: $$(-4)^2 = 16$$ Now, add and subtract 16 inside the parentheses: $$f(x) = (x^2 - 8x + 16 - 16) + 12$$ Then, combine these terms to form a perfect square trinomial: $$f(x) = ((x - 4)^2 - 16) + 12$$ Finally, simplify by combining constants: $$f(x) = (x - 4)^2 - 4$$

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form, $$f(x) = a(x - h)^2 + k$$ The axis of symmetry is given by the equation: $$x = h$$ From the vertex form we derived: $$f(x) = (x - 4)^2 - 4$$ The value of h is 4, so the axis of symmetry is: $$x = 4$$ This means every point on the parabola has a mirror counterpart across the line x = 4.

The minimum or maximum value of a quadratic function depends on the coefficient of the squared term (a). If a > 0, the parabola opens upwards and has a minimum value. If a < 0, it opens downwards and has a maximum value. In our function, $$f(x) = x^2 - 8x + 12$$ The coefficient of the squared term is +1, which is positive. Therefore, this function has a minimum value. The minimum value occurs at the vertex of the parabola. From the vertex form: $$f(x) = (x - 4)^2 - 4$$ The vertex is at $$(4, -4)$$ Thus, the minimum value of the function is -4, which is the y-coordinate of the vertex.

Graphing a quadratic function involves a few main steps. We already have the function in vertex form: $$f(x) = (x - 4)^2 - 4$$ 1. **Vertex & Axis of Symmetry:** Start by plotting the vertex at (4, -4). Draw the axis of symmetry, which is a vertical line through x = 4. 2. **Direction of Parabola:** Since the coefficient of x^2 is positive, the parabola opens upward. 3. **Additional Points:** Choose x-values around the vertex to determine other points on the graph. For instance: - For x = 3: $$f(3) = (3 - 4)^2 - 4 = 1 - 4 = -3$$ - For x = 5: $$f(5) = (5 - 4)^2 - 4 = 1 - 4 = -3$$ Plot the points (3, -3) and (5, -3). 4. **Drawing the Parabola:** Draw a smooth curve through these points and the vertex. Continue the curve upwards, symmetric relative to the axis of symmetry.