Short Answer 
The function f(x) = -x^2 + x + 6 is a downward-opening parabola with its vertex at (1/2, 11/4). The x-intercepts, found by solving the equation f(x) = 0, are at x = 2 and x = 3, which help in accurately sketching the graph.
Step 1: Understand the Shape of the Graph
The function f(x) = -x^2 + x + 6 represents a downward-opening parabola. This is determined by the negative coefficient of the x^2 term, indicating that the graph will peak at its vertex. The vertex is a critical point that represents the highest value of the graph.
Step 2: Calculate the Vertex
To find the vertex, use the vertex formula x = -b/(2a). Here, the values are:
- a = -1 (coefficient of x^2)
- b = 1 (coefficient of x)
Plugging these into the formula gives:
- x-coordinate: 1/2
By substituting this x value back into the function, we find that the vertex is at the point (1/2, 11/4).
Step 3: Find the X-Intercepts
The x-intercepts occur where the graph crosses the x-axis, meaning we set f(x) = 0. Solving the equation -x^2 + x + 6 = 0 using the quadratic formula yields:
- x = 2
- x = 3
These intercepts, along with the vertex, enable you to sketch the graph accurately, showing a downward parabola peaking at the vertex and crossing the x-axis at these two points.