Final answer:
The quadratic function f(x) = (x+1)(x-6) has an upward concavity, a y-intercept at (0, -6), x-intercepts at (-1, 0) and (6, 0), an axis of symmetry at x = 5/2, and a vertex at (5/2, -12.25).
Step-by-step explanation:
To analyze the quadratic function f(x) = (x+1)(x-6), we need to find its concavity, y-intercept, x-intercepts, the axis of symmetry, and the vertex.
Concavity:
A quadratic function f(x) = ax^2 + bx + c is concave up if a > 0 and concave down if a < 0. In the given function, the coefficient of the squared term is positive (a = 1), so the concavity is upward.
Y-Intercept:
The y-intercept occurs where x = 0. Substituting this into the function gives f(0) = (0+1)(0-6) = -6. Hence, the y-intercept is (0, -6).
X-Intercepts:
X-intercepts are the points at which the function crosses the x-axis, which happens when f(x) = 0. Setting the function to zero yields the points (x+1)(x-6) = 0, which translates into two x-intercepts of x = -1 and x = 6.
Axis of Symmetry:
The axis of symmetry for a quadratic function is given by the formula x = -b/2a. For the function f(x) = (x+1)(x-6) rewritten as x^2 - 5x - 6, the value of b is -5. Therefore, the axis of symmetry is x = 5/2.
Vertex:
The vertex of a quadratic function is the maximum or minimum point and lies on the axis of symmetry. Using the axis of symmetry x = 5/2, we can find the y-coordinate of the vertex by plugging this x-value into the function, which yields:
f(5/2) = ((5/2)+1)((5/2)-6) = 3.5*(-3.5) = -12.25. Hence, the vertex of the function is (5/2, -12.25)