993 10.1 Parabolas EXAMPLE 1 Graphing a Parabola (Horizontal Axis of Symmetry) Graph x + 3 = 1y - 222. Give the domain and range. SOLUTION The graph of x + 3 = 1y - 222, or x - 1-32 = 1y - 222, has vertex 1-3, 22 and opens to the right because a = 1, and 1 70. Plotting a few additional points gives the graph shown in Figure 3. S Now Try Exercise 9. x y 0 (1, 0) (1, 4) (–2, 1) (–3, 2) (–2, 3) y = 2 x + 3 = ( y – 2)2 –3 Figure 3 x y -3 2 -2 3 -2 1 1 4 1 0 The graph is symmetric with respect to its axis, y = 2. Domain: 3-3, ∞2 Range: 1-∞, ∞2 EXAMPLE 2 Graphing a Parabola (Horizontal Axis of Symmetry) Graph x = 2y2 + 6y + 5. Give the domain and range. GRAPHING CALCULATOR SOLUTION A horizontal parabola is not the graph of a function. To graph it using a graphing calculator in function mode, we must write two equations by solving for y. x - 1 2 = 2 ay + 3 2b 2 (*) from algebraic solution x - 0.5 = 21y + 1.522 Write with decimals. x - 0.5 2 = 1y + 1.522 Divide by 2. {Bx - 0.5 2 = y + 1.5 Take the square root on each side. y = -1.5{Bx - 0.5 2 Subtract 1.5, and rewrite. Figure 5 shows the graphs of the two functions defined in the final equation. Their union is the graph of x = 2y2 + 6y + 5. ALGEBRAIC SOLUTION x = 2y2 + 6y + 5 x = 21y2 + 3y 2 + 5 Factor out 2. x = 2 ay2 + 3y + 9 4 - 9 4b + 5 Complete the square; C 1 2 132D 2 = 9 4 . x = 2 ay2 + 3y + 9 4b + 2 a- 9 4b + 5 Distributive property x = 2 ay + 3 2b 2 + 1 2 Factor, and simplify. x - 1 2 = 2 ay + 3 2b 2 Subtract 1 2 . (*) The vertex of the parabola is A1 2 , - 3 2B. The axis is the horizontal line y = - 3 2 . Using the vertex and the axis and plotting a few additional points gives the graph in Figure 4. If we let y = 0, we find that the x-intercept is 15, 02, and because of symmetry, the point 15, -32 also lies on the graph. The domain is C 1 2 , ∞B, and the range is 1-∞, ∞2. x y (5, 0) (5, –3) 1 0 –1 ( ) , – 1 2 3 2 3 2 y =– x – = 2(y + )2 1 2 3 2 x = 2y2 + 6y + 5 Figure 4 −2 −6 2 8 y1 = −1.5 + Ä x−0.5 2 y2 = −1.5 − Ä x−0.5 2 S Now Try Exercise 17. Figure 5
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