Short Answer 
The function f(x) = (x Р2)(x Р6) is converted into vertex form f(x) = (x Р4)¬≤ Р4, identifying the vertex at (4, -4). The x-intercepts are (2, 0) and (6, 0), while the y-intercept is (0, 12), allowing for the complete graph of the parabola.
Step 1: Convert the Function to Vertex Form
Start with the given function f(x) = (x – 2)(x – 6). Expand the equation by distributing the terms:
- f(x) = x¬≤ Р6x Р2x + 12
- Now combine like terms: f(x) = x¬≤ Р8x + 12
- Next, complete the square to convert it into vertex form, leading to: f(x) = (x Р4)¬≤ Р4.
Step 2: Identify the Vertex of the Parabola
The vertex form of a parabola is expressed as g(x) = a(x Рh)¬≤ + k, where (h, k) represents the vertex. For our function, compare it with the vertex form:
- The vertex is (h, k) = (4, -4).
- This gives us the coordinate of the vertex of the function.
- Write down the vertex as: Vertex = (4, -4).
Step 3: Find the Intercepts and Graph the Function
To complete the graph, you’ll need to find the x-intercepts and y-intercept of the function. For the x-intercepts, set f(x) = 0 and solve:
- 0 = (x Р4)¬≤ Р4
- Solving gives x = 2 and x = 6, so x-intercepts are (2, 0) and (6, 0).
- To find the y-intercept, substitute x = 0: f(0) = 12, resulting in the y-intercept (0, 12).
Finally, plot the points (2, 0), (6, 0), and (0, 12) on a graph and connect them with a curve to visualize the parabola.