166 CHAPTER 3 Linear and Quadratic Functions SUMMARY Steps for Graphing a Quadratic Function fx ax bx c,a 0 2 ( ) = + + ≠ By Hand Option 1 Step 1 Complete the square in x to write the quadratic function in the vertex form f x a x h k. 2 ( ) ( ) = − + Step 2 Graph the function using transformations. Option 2 Step 1 Determine whether the parabola is concave up a 0 ( ) > or concave down a 0 . ( ) < Step 2 Find the vertex b a f b a 2 , 2 . ( ) ( ) − − Step 3 Find the axis of symmetry, x b a 2 . = − Step 4 Find the y -intercept, f 0( ) , and the x -intercepts, if any. • If b ac 4 0, 2 − > the graph of the quadratic function has two x-intercepts, which are found by solving the equation ax bx c 0. 2 + + = • If b ac 4 0, 2 − = the vertex is the x -intercept. • If b ac 4 0, 2 − < there are no x -intercepts. Step 5 Find an additional point using the y -intercept and the axis of symmetry. Step 6 Plot the points and draw the graph. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 3.3 Assess Your Understanding 1. Find the intercepts of the equation y x 9. 2 = − (pp. 20–21) 2. Find the real solutions of the equation x x 2 7 4 0. 2 + − = (pp. A47–A54) 3. To complete the square of x x5 , 2 − add the number . (p. A29) 4. To graph y x 4 , 2 ( ) = − shift the graph of y x2 = to the a distance of units. (pp. 112–115) 5. Find the discriminant of x x 2 5 8 0. 2 − − = Then identify the number of real solutions of the equation. (p. A52) 6. Complete the square of x x 3 7 . 2 + Factor the new expression. (pp. A29 and A50–A51) 7. Interactive Figure Exercise Exploring the Graph of the Quadratic Function Open the “Quadratic Functions” interactive figure, which is available in the Video & Resource Library in MyLab Math (under Sullivan Interactive Figures). (a) Set the values of a , h , and k as follows: a 2, = h k 3, 4. = = − What is the equation of the quadratic function? (b) Set the values of a , h , and k as follows: a 2, = h k 1, 3. = = − What is the vertex of the graph of the quadratic function? Express your answer as an ordered pair. (c) Set h to 1− and k to 1. Adjust the slider for a so that it scrolls from a 3 = − to a 3. = When the value of a 0, < the graph of the quadratic function is concave . (d) Use the applet to determine the quadratic function whose vertex is 1, 3 ( ) − and contains the point 2, 6. ( ) − 8. Interactive Figure Exercise Exploring the Discriminant Open the “Discriminant” interactive figure, which is available in the Video & Resource Library in MyLab Math (under Sullivan Interactive Figures). Be sure to check the “Discriminant” box. (a) Set the value of a to 1, − b to 1, − and c to 2. How many x -intercepts does the graph of the quadratic function have? (b) Set the value of a to 1, − b to 1, − and c to 2. Determine the value of the discriminant. (c) Set the value of a to 4, b to 4, and c to 1. How many x -intercepts does the graph of the quadratic function have? (d) Set the value of a to 4, b to 4, and c to 1. Determine the value of the discriminant. (e) Set the value of a to 2, − b to 2, and c to 1. − How many x -intercepts does the graph of the quadratic function have? Concepts and Vocabulary 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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