Lesson 16. Opening Exploration A Special Case

Transcription

1 Opening Exploration A Special Case 1. Consuela ran across the quadratic equation y = 4x 2 16 and wondered how it could be factored. She rewrote it as y = 4x 2 + 0x 16. A. Use one of the methods you ve learned to factor this quadratic function. B. What are the key features of the parabola s graph? x-intercepts:, y-intercept: vertex: (, ) C. Graph the quadratic in the grid at the right Unit 9: More with Quadratics Factored Form S.147

2 Factor the following quadratic functions. Use Consuela s idea of a 0 middle term if necessary. Look for patterns as you go. 2. y = x y = x What is the generic rule for factoring a quadratic in the form a 2 b 2? 5. The expression a 2 b 2 is called The Difference of Squares. Discuss with your partner where that name comes from. On of the other special cases is Perfect Square Trinomials. You ve already encountered a few of these but we ll focus on them now and see why they are special. 6. Factor each of the following. Use any method. A. y = x 2 + 6x + 9 B. y = x 2 4x A. What are the x-intercepts for these two equations? Remember the x-intercept is where y = 0. B. What is different about the x-intercepts for these two equations? Unit 9: More with Quadratics Factored Form S.148

3 8. Determine the key features of each quadratic and then graph the parabola. A. y = x 2 + 6x + 9 Standard Form B. y = x 2 4x + 4 Standard Form y = (x + )(x + ) Factored Form y = (x - )(x - ) Factored Form y = Vertex Form y = Vertex Form Key features: x-intercepts:, y-intercept: vertex: (, ) Key features: x-intercepts:, y-intercept: vertex: (, ) 9. Both of the quadratic equations in Exercise 8 are perfect square trinomials. What is special about their graphs? Unit 9: More with Quadratics Factored Form S.149

4 10. Let s look at a couple of perfect square trinomials that have a 1. Factor each one. A. y = 4x 2 + 4x + 1 B. y = 9x 2 12x Determine the key features of each quadratic and then graph the parabola. A. y = 4x 2 + 4x + 1 Standard Form B. y = 9x 2 12x + 4 Standard Form y = ( x + )( x + ) Factored Form y = Factored Form y = Vertex Form y = Vertex Form Key features: x-intercepts:, Key features: x-intercepts:, y-intercept: y-intercept: vertex: (, ) vertex: (, ) Unit 9: More with Quadratics Factored Form S.150

5 Practice Problems Factor each perfect square trinomial or difference of squares. Notice that these are not written as quadratic functions. They are simply expressions we could not graph them or tell their key features. 12. x 2 + 4x x 2 20x x 2 + 2x x x x x x 2 20x x 2 8x x x x x x x 2 49 Unit 9: More with Quadratics Factored Form S.151

6 Lesson Summary Difference of Squares (ax) 2 b 2 (ax b) (ax + b) Perfect Square Trinomials (ax) 2 + 2abx + b 2 (ax) 2 2abx + b 2 (ax + b) 2 (ax b) 2 Unit 9: More with Quadratics Factored Form S.152

7 Homework Problem Set Factor the following examples of the difference of perfect squares. Notice that these are not written as quadratic functions. They are simply expressions we could not graph them or tell their key features. 1. tt xx h 2 36kk bb 2 5. xx xx yy 2 100zz 2 8. aa 4 bb 6 9. Challenge rr 4 16ss 4 (Hint: This one factors twice.) Unit 9: More with Quadratics Factored Form S.153

8 10. For each of the following, factor out the greatest common factor. a. 6yy b. 27yy yy c. 21bb 15aa d. 14cc 2 + 2cc e. 3xx The measure of a side of a square is xx units. A new square is formed with each side 6 units longer than the original square s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and count the squares and rectangles.) Original Square xx Unit 9: More with Quadratics Factored Form S.154

9 12. In the accompanying diagram, the width of the inner rectangle is represented by xx 3 and the length by xx + 3. The width of the outer rectangle is represented by 3xx 4 and the length by 3xx + 4. a. Write an expression to represent the area of the larger rectangle. b. Write an expression to represent the area of the smaller rectangle. MIXED REVIEW Factor completely xx 2 25xx 14. 9xx xx 2 30xx xx 2 + 7xx xx 2 + 7xx xx xx xx xx xx 2 + 4xx + 1 Unit 9: More with Quadratics Factored Form S.155

10 CHALLENGE PROBLEMS 21. The area of the rectangle at the right is represented by the expression 1111xx xx + 22 square units. Write two expressions to represent the dimensions, if the length is known to be twice the width. 1111xx xx Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and has the same dimensions. They agree that each has an area of 2xx 2 + 3xx + 1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing does he need? Write your answer as an expression in xx. Note: This question has two correct approaches and two different correct solutions. Can you find them both? Unit 9: More with Quadratics Factored Form S.156

11 Spiral Review Factoring Lessons Factor the following quadratic expressions xx xx xx 2 + 5xx x 2 12x x 2 21x xx 2 xx xx 2 + 7xx x 2 2x x 2 + 2x x x x x + 4 Unit 9: More with Quadratics Factored Form S.157

12 Unit 9: More with Quadratics Factored Form S.158