Solution The only difference between the two functions is the value of c . Since the value of c is increased by 4, the graph of ( ) g x can be found by vertically shifting the graph of ( ) f x up 4 units. In order to compare two quadratic functions where the values of a , b , and/or c change, it is easier to look at the standard form of a quadratic function. By completing the square we get 2 2 2 2 2 2 2 2 2 2 2 2 4 2 4 2 ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ( )) ( ) ( ) b b b b a a a a b b b b a a a a f x ax bx c a x x c a x x c a a x c a x c a x h k where 2 b a h and 2 4 ( ) b a k c f h . In standard form, the graph of the parabola can be found by shifting the graph of 2 ( ) f x x horizontally (right if 0 h , left if 0 h ) by h units, vertically expanding or contracting by a factor of a , vertically reflecting if 0 a , and then shifting vertically by k units. Notice that by performing this transformation, the graph of a quadratic in standard form has a vertex at the point ( , ) h k . Table 1.1 Summary Chart for Parabolas Let 2 ( ) ( ) f x a x h k . Comparing this to the parabola 2 x , 1. If 0 a then ( ) f x opens upward and the vertex is the lowest point. 2. If 0 a then ( ) f x vertically reflected and the vertex is the highest point. 3. If 0 1 a then ( ) f x is vertically compressed relative to 2 x . 4. If 1 a then ( ) f x is vertically expanded relative to 2 x . 5. If 0 k , then ( ) f x is vertically shifted upwards by k units. 6. If 0 k , then ( ) f x is vertically shifted downwards by k units. 7. If 0 h , then ( ) f x is horizontally shifted to the right by h units. 8. If 0 h , then ( ) f x is horizontally shifted to the left by h units. 9. The vertex is at ( , ) h k . Example 1.13 Find the vertex of the parabola 2 ( ) 2 3 6 f x x x and determine whether it is a maximum or minimum. Solution The x -coordinate of the vertex is given by ( 3) 0.75 2 2(2) b x a The y -coordinate of the vertex can be found by plugging in the corresponding x value: 2 (0.75) 2(0.75) 3(0.75) 6 4.875 f
Therefore, the vertex of the parabola is located at (0.75, 4.875). Since 2 0 a , then the parabola opens upward from the vertex, so the vertex is a minimum. Example 1.14 How is the graph of 2 1 5 ( ) ( 2) 4 f x x different from the graph of 2 ( ) g x x ? Solution The graph will be horizontally shifted to the left by 2 units, reflected over the x -axis, vertically contracted by a factor of 1 5 , and then vertically shifted down 4 units. Using the Plotting Applet, these transformations can be seen in Figure 1.14. Figure 1.13 Graphs for Example 1.11 using the Plotting Applet Many real-life situations are quadratic in nature, as they portray information that increases to a point and then decreases, such as revenue or profit, or decreases to a point and then increases, such as costs.
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- Fall '16
- Supply And Demand, Quadratic equation