Sketch the graph of f (x) x2, then move every point on the graph of f right 2 and down 1 to obtain the graph of g(x). This moves the vertex from (0, 0) to (2, 1). Notice that the axis of symmetry of g(x) moves 2 units to the right also. Its equation is x 2. The domain of g(x) is (q, q); the range is 1, q). 2 Graph f(x) a(x h)2 k Using Characteristics f (x) x2 5 g(x) (x 2)2 1 Axis of symmetry x y Vertex (0, 0) Axis of symmetry of f(x) 5 5 5 Vertex (2, 1) of g(x) of a Parabola When a quadratic function is in the form f (x) a(x h)2 k, we can read the vertex directly from the equation. Furthermore, the value of a tells us if the parabola opens upward or downward and whether the graph is narrower, wider, or the same width as y x2. Procedure Graphing a Quadratic Function of the Form f(x) a(x h)2 k 1) The vertex of the parabola is (h, k). 2) The axis of symmetry is the vertical line with equation x h. 3) If a is positive, the parabola opens upward. If a is negative, the parabola opens downward. 4) If 0a 0 1, then the graph of f (x) a(x h)2 k is wider than the graph of y x2. If 0a 0 1, then the graph of f (x) a(x h)2 k is narrower than the graph of y x2. If a 1 or a 1, the graph is the same width as y x2. EXAMPLE 5 Graph f (x) 2(x 1)2 4. Also fi nd the x- and y-intercepts. Solution Here is the information we can get from the equation: 1) h 1 and k 4. The vertex is (1, 4). 2) The axis of symmetry is x 1. 3) a 2. Because a is positive, the parabola opens upward. 4) Since 0a 0 1, the graph of f (x) 2(x 1)2 4 is narrower than the graph of f (x) x2. www.mhhe.com/messersmith SECTION 10.5 Quadratic Functions and Their Graphs 657
messersmith_power_intermediate_algebra_1e_ch4_7_10
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