Factored Form When a = 1
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1 Lesson 4 Hart Interactive Algebra Lesson 4: Factored Form When a = Opening Activity Graph Exchange Your group will need: one quadratic graph. A. For your given graph, circle the graph number on the table below and determine the x-intercepts. Then write the equation of the graph in factored form, standard form and vertex form. You can assume that for each graph there is no stretching or shrinking so a = in all cases, where ax2 + bx + c = y is the equation of the quadratic function. Be prepared to share out your results with the class. Graph Number xintercepts Factored Form Standard Form Vertex Form B. Once you have finished your graph, you ll be given a new graph from another group. Fill in the chart with the x-intercepts and equations. Continue getting new graphs from other groups until you have seen all 5 graphs. Sharing Your Results Math Practice 3 2. Each group will share their x-intercepts and equations with the class. If your group s equations don t match the presenting group s equations, be sure to point it out. It is important that all equations are correct for the next part of this lesson. Lesson 4: Unit : Factored Form When a = More with Quadratics Factored Form This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE S.29
2 Hart Interactive Algebra Lesson 4 Looking for Patterns 3. Look back at the factored forms and standard forms of the graph. Discuss with your group what relationships you see in the numbers. List 3 patterns you see between the two forms of the equation. 4. If we focus on the numbers, we may be able to see the patterns a little more clearly. We can write the equations abstractly as y = (x + p)( x + q) (factored form) and y = ax 2 + bx + c (standard form). Use your equations from Exercise to fill in the chart. Then write the pattern you see between the p and q values and the a, b and c values. Graph has been started for you. Factored Form Standard Form p q a b c Pattern For the equation y = x 2 + 7x + 2, what would the factored form would be? y = (x + )(x + ) Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.30 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
3 Hart Interactive Algebra Lesson 4 6. In Exercise 4, both x-intercepts were negative. Now we ll look at x-intercepts that are positive. For each graph below, write the factored form of the equation and the standard form of the equation. Then write the p, q, b, and c values. What pattern do you see? The first one has been started for you. Graph Factored Form Standard Form p q a b c Patterns y = (x )(x 5) y= x 2 6x Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.3 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
4 Hart Interactive Algebra Lesson 4 7. For the equation y = x 2 9x + 4, what would the factored form be? Why must both signs in the factored form be negative? y = (x - )(x - ) Exploration Negative and Positive x-intercepts 8. In all of these cases, both x-intercepts were negative or both were positive. It is possible to have a factored form with one negative x-intercept and one positive x-intercept. A. Create an equation in factored form with one negative x-intercept and one positive x-intercept. So that you can graph your equation, keep your x-intercepts between -5 and +5. What are the x-intercepts? Equation: x-intercepts: and B. Rewrite your equation in standard form. What is the y-intercept? C. What is the axis of symmetry of your quadratic? D. What is the vertex of your quadratic? E. Graph your quadratic equation. F. What is the equation of your quadratic in vertex form? For the equation y = x 2 + 5x 24, what would the factored form be? Be sure to write in the signs. y = (x )(x ) 0. For the equation y = x 2 5x 24, what would the factored form be? Be sure to write in the signs. y = (x )(x ) Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.32 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
5 Hart Interactive Algebra Lesson 4 Practice Problems For each equation below, write the factored form of the equation.. y = x 2 + 4x y = x 2 + 5x y = x 2 4x y = x 2 + 5x y = x 2 7x y = x 2 + 7x y = x 2 x 2 8. y = x 2 + 4x 2 9. y = x 2 x y = x 2 2x 8 2. y = x 2 + 2x y = x 2 7x 8 Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.33 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
6 Hart Interactive Algebra Lesson 4 Lesson Summary Factoring trinomials means finding two binomials that when multiplied together produce the given trinomial. For trinomials of the form y = ax 2 + bx + c and a =, of c add up to. In factored form, y = (x + p)(x + q), p q = and p + q =. Some techniques to factoring include the X method as shown below. Example: y = x 2 + 7x + 0 = (x + 2)(x + 5) a c p q p 0 q b Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.34 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
7 Hart Interactive Algebra Lesson 4 Homework Problem Set For each equation below, write the factored form of the equation.. y = x 2 + 9x y = x 2 + 8x y = x 2 + 3x y = x 2 + 0x y = x 2 + x y = x 2 + 6x y = x 2 8x y = x 2 9x y = x 2 8x y = x 2 4x 60. y = x 2 + x y = x 2 2x 5 Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.35 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
8 Hart Interactive Algebra Lesson 4 3. The expression x 2 + 3x + c can be factored and c is an integer. A. What is a value c can have? Show how it would be factored. B. Are there any other values of c that would make the expression factorable? Explain your reasoning. 4. The expression x 2 + bx + 24 can be factored and b is an integer. A. What is a value b can have? Show how it would be factored. B. Are there any other values of b that would make the expression factorable? Explain your reasoning. Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.36 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
9 Hart Interactive Algebra Lesson 4 For each algebra tile model below, write the expression it represents as a product and as a sum Product: Product: Product: Sum: Sum: Sum: 8. Factor each of the following expressions. You may draw a model. A. 4x 2 8x B. 9x x C. 0x 2 25x 9. Discuss how you could factor the expression 2x The algebra tile model is shown on the right. Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.37 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
10 Hart Interactive Algebra Lesson 4 Lesson 4: Factored Form When a = Unit : More with Quadratics Factored Form S.38 This work is derived from Eureka Math and licensed by Great Minds. 205 Great Minds. eureka-math.org This file derived from ALG I--TE
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