325 3.1 Quadratic Functions and Models Parabolas are symmetric with respect to a line (the y-axis in Figure 1). This line is the axis of symmetry, or axis, of the parabola. The point where the axis intersects the parabola is the vertex of the parabola. As Figure 2 shows, the vertex of a parabola that opens down is the highest point of the graph, and the vertex of a parabola that opens up is the lowest point of the graph. Opens up Axis Axis Opens down Vertex Vertex Figure 2 Graphing Techniques Graphing techniques may be applied to the graph of ƒ1x2 = x2 to give the graph of a different quadratic function. Compared to the basic graph of ƒ1x2 = x2, the graph of F1x2 = a1x - h22 + k, with a≠0, has the following characteristics. F1x2 =a1x −h22 +k • Opens up if a 70 • Opens down if a 60 • Vertically stretched (narrower) if a 71 • Vertically shrunk (wider) if 0 6 a 61 Horizontal shift: • Right h units if h 70 • Left h units if h 60 Vertical shift: • Up k units if k 70 • Down k units if k 60 EXAMPLE 1 Graphing Quadratic Functions Graph each function. Give the domain and range. (a) ƒ1x2 = x2 - 4x - 2 (by plotting points) (b) g1x2 = - 1 2 x 2 1and compare to y = x2 and y = 1 2 x 22 (c) F1x2 = - 1 2 1x - 422 + 3 (and compare to the graph in part (b)) SOLUTION (a) See the table with Figure 3. The domain of ƒ1x2 = x2 - 4x - 2 is 1-∞, ∞2, the range is 3-6, ∞2, the vertex is the point 12, -62, and the axis has equation x = 2. Figure 4 shows how a graphing calculator displays this graph. 2 x = 2 –6 –2 3 x y 0 f(x) = x2 – 4x – 2 (2, –6) Figure 3 f(x) = x2 − 4x−2 −10 −10 10 10 Figure 4 x ƒ1x2 -1 3 0 -2 1 -5 2 -6 3 -5 4 -2 5 3 (b) Think of g1x2 = - 1 2 x 2 as g1x2 = - A1 2 x 2B. The graph of y = 1 2 x 2 is a wider version of the graph of y = x2, and the graph of g1x2 = - A1 2 x 2B is a reflection of the graph of y = 1 2 x 2 across the x-axis. See Figure 5 on the next page. The vertex is the point 10, 02, and the axis of the parabola is the line x = 0 (the y-axis). The domain is 1-∞, ∞2, and the range is 1-∞, 04.
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