Lesson 19. Opening Discussion

Transcription

1 Opening Discussion 1. Think about the forms of the quadratic equations you ve written throughout this module. We have gone from vertex form to standard form and from factored form to standard form. Draw arrows between any other forms that are relatively easy to change from one to the other. An example equation is given for each one to help you think about the forms. Unit 12: Completing the Square & The Quadratic Formula S.137

2 2. When equations are in the form, y = (x h) 2 + k, what is the vertex? 3. In Lesson 18, you learned to complete the square for quadratic expressions. We ll now turn our attention to quadratic equations. What is the difference between an expression and an equation? 4. Sergio thought about how to find the vertex form from standard form and wrote the following problem. Rolando looked at Sergio s work and had some questions. Use Rolando s thought bubbles to help explain the steps Sergio took. y = x x + 9 Why did Sergio write (x + 5)? How did he know 5 is the h value? Why must k equal 9? y = (x + 5) 2 + k k = k = 9 k = -16 y = (x + 5) 2-16 How can I check that this is equivalent to the original equation? Complete the square for each quadratic equation below. Then determine the vertex of each function. 5. y = x x y = x 2 2x 8 7. y = x 2 + 6x 5 y = (x ) 2 + y = (x ) 2 + y = (x ) 2 + vertex: (, ) vertex: (, ) vertex: (, ) Unit 12: Completing the Square & The Quadratic Formula S.138

3 8. Logan was given the following equation and asked to write it in vertex form. Where is Logan s mistake? What is the correct answer? y = x 2 6x + 13 y = (x + 3) 2 + k k = k = 13 k = 4 y = (x + 3) Practice Exercises - Circle the mistake in each problem and then fix the error to find the correct vertex form of the quadratic function. 9. y = x 2 6x y = x 2 + 2x y = x x + 35 y = (x - 3) 2 + k k = k = 5 k = 14 y = (x - 3) y = (x - 1) 2 + k k = k = -8 k = -9 y = (x - 1) 2 9 Fix the Error: Fix the Error: Fix the Error: y = x 2 + 5x + 7x + 35 y = x(x + 5) + 7(x + 5) y = (x + 5)(x + 7) Unit 12: Completing the Square & The Quadratic Formula S.139

4 Connecting to Geometry For each figure below, write an equation for the area. Then rewrite each equation in vertex form and determine the vertex. 12. Square 13. Rectangle 14. Triangle A = (x ) 2 + A = (x ) 2 + A = (x ) 2 + Vertex: (, ) Vertex: (, ) Vertex: (, ) Discussion 15. A. If you knew the area of the square in Exercise 12 was 25 square feet, how could you determine the value of x? B. Suppose the area of the square was 1 square inch, what value would x have? C. Is it possible for x to be 3, in Exercise 12? Explain. Unit 12: Completing the Square & The Quadratic Formula S.140

5 Homework Problem Set Write each quadratic equation in vertex form. 1. y = x x 3 2. y = x 2 6x y = x 2 4x 5 4. y = x 2 + 2x y = x 2 8x 2 6. y = x 2 + 7x 3 7. y = x 2 5x y = x 2 3x y = x x y = x 2 6x 11. y = x x 12. y = x 2 4 Unit 12: Completing the Square & The Quadratic Formula S.141

6 Circle the mistake in each problem and then fix the error to find the correct vertex form of the quadratic function. 13. y = x x y = x y = x 2 + 8x y = (x + 15) 2 + k k = k = -2 k = -227 y = (x +15) y = x 2 3x + 3x 9 y = x(x 3) + 3(x 3) y = (x + 3)(x 3) Fix the Error: Fix the Error: Fix the Error: y = (x 4) 2 + k k = k = 0 k = 16 y = (x 4) Challenge 16. The quadratic function y = (x + 3)(x + 5) can be modeled with the figure at the right. A. Write the equation in vertex form. x 2 x x x B. Draw a new model that uses the same number of rectangles and squares to show that the vertex model is equivalent to the factored model shown. Unit 12: Completing the Square & The Quadratic Formula S.142