Answer:
1) The equation is given in factored form
2) The key characteristics of the parabola are;
a. The parabola has 2 real roots and extends to infinity on both sides
b. The x-intercepts of the parabola are (1/2, 0) and (-6, 0)
c. The y intercept is (0, -6)
d. The vertex point is (-2.75, -21.125)
Explanation:
The given equation is f(x) = (2·x -1)(x + 6)
Therefore the equation is given in factored form
The key characteristic of the parabola revealed from the form f(x) = (2·x -1)(x + 6) are;
1) Writing the parabola in the intercept form, a(x - p)(x -q) we have;
f(x) = (2·x -1)(x + 6) = 2·(x - 1/2)(x + 6)
p = 1/2, q = -6
Therefore;
Given that a is positive the parabola is concave upwards
2) a. The parabola has 2 real roots and extends to infinity on both sides
b. The x-intercepts of the parabola are p and q which are (1/2, 0) and (-6, 0)
c. The y intercept is a·(-p)·(-q) = 2×(-1/2)×6 = -6
The y intercept is (0, -6)
d. Expanding we get;
f(x) = 2·x² + 11·x - 6
From the
The vertex is h = -b/(2·a) = -11/(2×2) = -2.75
h(2.75) = 2×(-2.75)² + 11×(-2.75) - 6 = -21.125
Therefore, the function is symmetrical about the vertex point (-2.75, -21.125)