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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 parabolaâ€™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.

The Graph of f (x) = ax2

Parabola

6

5

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3

2

1

The Graph of f (x) = a(x â€“ 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 â€“ 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 â€“ h)2 + k

f (x) = 2(x + 3)2- 5

Graphing f (x) = a(x â€“ h)2 + k

The graph of f (x) = a(x â€“ h)2 + k has the same shape

as the graph of y = a(x â€“ h)2.

If kis positive, the graph of y = a(x â€“ h)2is

shifted k units up.

If kis negative, the graph of y = a(x â€“ h)2is

shifted |k| units down.

The vertex is (h, k), and the axis of symmetry is x = h.

The domain of f is (ï€ï‚¥,ï‚¥).

If a> 0, the range is f is [k,ï‚¥). A minimum function value is k, which occurs when x = h.

For a< 0, the range of f is (ï€ï‚¥, k]. A maximum function value of koccurs when x = h.