The graph of a quadratic function of the form f(x) = a x 2 + b x + c
is a parabola
Properties of Graphs of Quadratic Functions.
a) If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.
b) As | a| increases, the parabola becomes narrower; as | a| decreases, the parabola becomes wider.
c) The lowest point of a parabola (when a > 0) or the highest point (when a < 0) is called the vertex.
d) The domain of a quadratic function is R, because the graph extends indefinitely to the right and to the left. If (h, k) is the vertex of the parabola, then the range of the function is [k,+ ∞ ) when a > 0 and (- ∞, k] when a < 0.
e) The graph of a quadratic function is symmetric with respect to a vertical line containing the vertex. This line is called the axis of symmetry. If (h, k) is the vertex of a parabola, then the equation of the axis of symmetry is x = h.
How to calculate the vertex of a parabola
1. To determine the vertex of the graph of a quadratic function, f(x) = ax2+ bx + c, we can either do it:
a) Completing the square to rewrite the function in the form f(x) = a(x – h)2 + k. The vertex is (h, k).
b) Using the formula to find the x-coordinate of the vertex and then, the y- coordinate of the vertex can be determined by evaluating . The vertex is (x,y).
How to calculate the intercepts of a parabola.
To find the y-intercept of the parabola, find f(0); to find the x-intercepts, solve the quadratic equation ax2+ bx + c = 0.