The Graph of f ( x ) = ax 2

2. The Graph of f (x) = ax2 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.

3. The Graph of f (x) = ax2 Parabola

4. 6 5 4 3 2 1

6. The Graph of f (x) = a(x – h)2 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.

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

15. 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 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.