162 CHAPTER 3 Linear and Quadratic Functions Figure 19 ( ) =− + − f x x x6 5 2 Axis of symmetry x 5 3 x y –4 4 7 (3, 4) (5, 0) (6, –5) (0, –5) (1, 0) 4 Step-by-Step-Solution Step 1 Determine whether the graph of f is concave up or concave down. Step 2 Determine the vertex and axis of symmetry of the graph of f . How to Graph a Quadratic Function by Hand Using Its Properties Graph f x x x6 5 2 ( ) = − + − using its properties. Find the domain and the range of f . Determine where f is increasing and where it is decreasing. Determine where f x 0 ( ) > and where f x 0. ( ) < EXAMPLE 3 Step 3 Determine the intercepts of the graph of f . Step 4 Use the information in Steps 1 through 3 to graph f . For f x x x6 5, 2 ( ) = − + − note that a b 1, 6, = − = and c 5. = − Because a 1 0, = − < the parabola is concave down. The x-coordinate of the vertex is h b a2 6 2 1 3 ( ) = − = − − = The y-coordinate of the vertex is k f 3 3 6 3 5 9 18 5 4 2 ( ) = =−+⋅−=−+ −= The vertex is 3, 4 . ( ) The axis of symmetry is x 3. = The y-intercept is f 0 5. ( ) = − The x-intercepts are found by solving f x 0. ( ) = This results in the equation x x6 5 0, 2 − + − = which we can solve by factoring: x x x x x x x x 6 5 0 1 5 0 1 0 or 5 0 1 or 5 2 ( )( ) − + − = − − − = − = − = = = The x-intercepts are 1 and 5. Factor. Use the Zero-Product Property. The graph is illustrated in Figure 19. Notice that we used the y-intercept and the axis of symmetry, x 3, = to obtain the additional point 6, 5 ( ) − on the graph. The domain of f is the set of all real numbers. Based on the graph, the range of f is the interval , 4 . ( ] −∞ The function f is increasing on the interval , 3 ( ] −∞ and decreasing on the interval 3, . [ )∞ Note that f x 0 ( ) > where the graph of f is above the x-axis, and f x 0 ( ) < where the graph of f is below the x-axis. So, f x 0 ( ) > on the interval 1, 5 ( ) or for x 1 5, < < and f x 0 ( ) < on the interval , 1 5, ( ) ( ) −∞ ∪ ∞ or for x x 1, 5. < > Graph the function in Example 3 by completing the square and using transformations. Which method do you prefer? Now Work PROBLEMS 29 AND 45(b)—(f) If the graph of a quadratic function has only one x-intercept or no x-intercepts, it is usually necessary to plot an additional point to obtain the graph. Graphing a Quadratic Function Using Its Vertex, Axis, and Intercepts (a) Graph f x x x6 9 2 ( ) = − + by determining whether the parabola is concave up or concave down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. (b) Find the domain and the range of f . (c) Determine where f is increasing and where it is decreasing. (d) Determine where f x 0 ( ) > and where f x 0. ( ) < EXAMPLE 4
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