The Graph of f (x) = ax 2 All quadratic functions have graphs similar to y = x 2. 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) = x 2, the y-axis is the axis of symmetry. The point (0, 0) is known as the vertex of this parabola.

Parabola The Graph of f (x) = ax 2

The Graph of f (x) = a(x – h) 2 We could next consider graphs of f (x) = ax 2 + 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.

The Graph of f (x) = a(x – h) 2 + k f (x) = 2(x + 3) 2  5

Graphing f (x) = a(x – h) 2 + k The graph of f (x) = a(x – h) 2 + k has the same shape as the graph of y = a(x – h) 2. If k is positive, the graph of y = a(x – h) 2 is shifted k units up. If k is negative, the graph of y = a(x – h) 2 is 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 k occurs when x = h.