The Graph of f ( x ) = ax 2.
All quadratic functions have graphs similar to y = x2. Such curves are called parabolas. They are U-shaped and symmetric with respect to a vertical line known as the parabola’s axis of symmetry. For the graph of f (x) = x2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.
We could next consider graphs of f(x) = ax2 + bx + c, where b and c are not both 0. It turns out to be convenient to first graph f(x) = a(x – h)2, where h is some constant. This allows us to observe similarities to the graphs drawn in previous slides.
f (x) = 2(x + 3)2- 5
The graph of f (x) = a(x – h)2 + k has the same shape
as the graph of y = a(x – h)2.
If kis positive, the graph of y = a(x – h)2is
shifted k units up.
If kis negative, the graph of y = a(x – h)2is
shifted |k| units down.
The vertex is (h, k), and the axis of symmetry is x = h.
The domain of f is (,).
If a> 0, the range is f is [k,). A minimum function value is k, which occurs when x = h.
For a< 0, the range of f is (, k]. A maximum function value of koccurs when x = h.