Investigation. So far you have worked with quadratic equations in vertex form and general. Getting to the Root of the Matter. LESSON 9.


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1 DA2SE_73_09.qd 0/8/0 :3 Page Factored Form LESSON 9.4 So far you have worked with quadratic equations in verte form and general form. This lesson will introduce you to another form of quadratic equation, the factored form: Mathematicians assume the right to choose, within the limits of logical contradiction, what path they please in reaching their results. y a r r2 This form helps you identify the roots, r and r2, of an equation. In the investigation you ll discover connections between the equation in factored form and its graph. You ll also use rectangle diagrams to convert the factored form to the general form and vice versa. Then in the eample you ll learn how to use a special property to find the roots of an equation. HENRY ADAMS Investigation Getting to the Root of the Matter First you ll find the roots of an equation in factored form from its graph. You will need graph paper Step On your calculator, graph the equations y 3 and y 4 at the same time. Step 2 What is the intercept of each equation you graphed in Step? Step 3 Graph y ( 3)( 4) on the same set of aes as before. Describe the graph. Where are the intercepts of this graph? Step 4 Epand y ( 3)( 4) to general form. Graph the equation in general form on the same set of aes. What do you notice about this parabola and its intercepts? Is the graph of y ( 3)( 4) a parabola? LESSON 9.4 Factored Form
2 Step Step Step 7 Step 8 Now you ll learn how to find the roots from the general form. Complete the rectangle diagram whose sum is 2. A few parts on the diagram have been labeled to get you started. Write the multiplication epression of the rectangle diagram in factored form. Use a graph or table to check that this form is equivalent to the original epression. Find the roots of the equation 0 2 from its factored form. Rewrite each equation in factored form by completing a rectangle diagram. Then find the roots of each. Check your work by making a graph. a b. 0 2 c d EXAMPLE A Now you have learned three forms of a quadratic equation. You can enter each of these forms into your calculator to check that they are equivalent. Here are three equivalent equations that describe the height in meters, y, of an object in motion for seconds after being thrown upward. Each equation gives different information about the object. Ver te form y 4.9(.7) 2.87 General form y Factored form y 4.9( 0.)( 3.) Which form is best? The answer depends on what you want to know. The verte form tells you the maimum height and when it occurs in this case,.87 meters after.7 seconds (the verte). The general form tells you that the object started at a height of.7 meters (the yintercept). The coefficients of and 2 give some information about the starting velocity and acceleration. The factored form tells you the times at which the object s height is zero (the roots). You have already learned how to convert to and from the general form of a quadratic equation. Eample A will show you how to get the verte form from the factored form. y Write the equation for this parabola in verte form, factored form, and general form. 9 Solution From the graph you can see that the intercepts are 3 and. So the factored form contains the binomial epressions ( 3) and ( ). 9 (, 0) (3, 0) 9 (, 8) 9 CHAPTER 9 Quadratic Models
3 If you graph y ( 3)( ) on your calculator, you ll see it has the same intercepts as the graph shown here, but a different verte. The new verte is (, ). y y ( 3)( ) (, ) 20 The new verte needs to be closer to the ais, so you need to find the vertical shrink factor a. The original verte of the graph shown is (, 8). So the graph of the function must 8 have a vertical shrink by a factor of, or 0.. The factored form is y 0.( 3)( ). A calculator graph of this equation looks like the desired parabola. Now you know that the value of a is 0. and that the verte is (, 8). Substitute this information into the verte form to get y 0.( ) 2 8. Epand either form to find the general form. Y =.(X 3)(X + ) X = Y = 8 [,,, 0, 0, ] y 0.( 3)( ) Original equations. y 0.( ) 2 8 y Epand using rectangle y diagrams y Combine like terms. y y Distribute and combine. y So the three forms of the quadratic equation are Ver te form y 0.( ) 2 8 General form y Factored form y 0.( 3)( ) LESSON 9.4 Factored Form 7
4 keymath.com/da EXAMPLE B Solution When finding roots it is helpful to use the factored form. In Eample A, one root is 3 because 3 is the value that makes ( 3) equal to 0. The other root is because it makes ( ) equal to 0. Think of numbers that multiply to zero. If ab 0 or abc 0, the zeroproduct property tells you that a, or b, or c must be 0. In an equation like ( 3)( ) 0, at least one of the factors must be zero. The roots of an equation are sometimes called the zeros of a function because they make the value of the function equal to zero. [ You can further eplore the relationship between factored form, roots, and intercepts using the Dynamic Algebra Eploration at ] The ability to factor polynomials is also useful when simplifying rational epressions, as you ll see in the net eample. When a polynomial equation is in factored form, and reduced, it s much easier to predict or identify characteristics such as intercepts and asymptotes. A rational epression can be reduced if there is a common factor in both the numerator and denominator. Reduce the epression by factoring. Then check your answer with a graph. First factor the quadratic epressions in the numerator and denominator. Using a rectangle diagram may help. Numerator: Denominator: Find two numbers whose product is 24 and sum is 2. 2 Find two numbers whose product is and sum is So, ( )( 4) and 2 7 ( )( ). Now reduce the rational epression. Be sure to state any restrictions on the variable ( ) ( 4) 7 ( ) ( ) ( ) ( 4) ( ) ( ) 4, where and You can check your work by graphing y and y 4 7. Ifthe epressions are equivalent the graphs should be the same, ecept for any points that may be undefined in one graph, but defined in the other. 8 CHAPTER 9 Quadratic Models
5 y y f() g() The only difference in the graphs is that the first graph is missing a point at (, 2). So the reduced epression is equivalent to the original epression. EXERCISES Practice Your Skills You will need your graphing calculator for Eercises 2, 3, 9, 2, and.. Use the zeroproduct property to solve each equation. a. ( 4)( 3.) 0 a b. 2( 2)( ) 0 c. ( 3)( 7)( 8) 0 d. ( 9)( 3) 0 2. Graph each equation and then rewrite it in factored form. a. y a b. y 2 24 c. y d. y Name the intercepts of the parabola described by each quadratic equation. Then check your answers with a graph. a. y ( 7)( 2) a b. y 2( )( 8) c. y 3( )( 7) d. y (0.4 2)( 9) 4. Write an equation of a quadratic function that corresponds to each pair of intercepts. Assume there is no vertical stretch or shrink. a. 2. and a b. 4 and 4 c. 2 and 2 d. r and r 2. Consider the equation y ( )( 3). a. How many intercepts does the graph have? b. Find the verte of this parabola. c. Write the equation in verte form. Describe the transformations of the parent function, y 2. Reason and Apply. Is the epression on the left equivalent to the epression on the right? If not, change the right side to make it equivalent. a ? ( 3)( 4) b. 2 30? ( )( ) c ? (2 7)( ) a d ? ( ) 2 e. 2 2? ( )( ) f. 2 3? ( ) 2 LESSON 9.4 Factored Form 9
6 7. Use a rectangle diagram to factor each epression. a. 2 7 a b c a d e a f The sum and product of the roots of a quadratic equation are related to b and c in y 2 b c. The first row in the table below will help you to recognize this relationship. a. Complete the table. Factored form Roots Sum of roots Product of roots General form y ( 3)( 4) 3 and (3)(4) 2 y 2 2 and 2 y ( )( ) 0 2 b. Use the values of b and c to find the roots of MiniInvestigation In this eercise you will discover whether knowing the intercepts determines a unique quadratic equation. Work through the steps in 9a e to find an answer. Graph each equation to check your work. a. Write an equation of a parabola with intercepts at 3 and 7. b. Name the verte of the parabola in 9a. c. Modify your equation in 9a so that the graph is reflected across the ais. Where are the intercepts? Where is the verte? d. Modify your equation in 9a to apply a vertical stretch with a factor of 2. Where are the intercepts? Where is the verte? a e. How many quadratic equations do you think there are with intercepts at 3 and 7? How are they related to one another? 0. Write a quadratic equation of a parabola with intercepts at 3 and 9 and verte at (3, 9). Epress your answer in factored form. a. APPLICATION The school ecology club wants to fence in an area along the riverbank to River protect endangered wildflowers that grow there. The club has enough money to buy 200 feet of fencing. It decides to enclose a rectangular space. The fence will form three sides of the rectangle, and the riverbank will Length form the fourth side. a. If the width of the enclosure is 30 feet, how much fencing material is available for the length? Sketch this situation. What is the area? a b. If the width is w feet, how much fencing material remains for the length, l? c. Use your answer from b to write an equation for the area of the rectangle in factored form. Check your equation with your width and area from a. d. Which two different widths would give an area equal to 0? a e. Which width will give the maimum area? What is that area? Width 20 CHAPTER 9 Quadratic Models
7 DA2SE_73_09.qd 0/8/0 :4 Page 2 2. MiniInvestigation Consider the equation y 2 9. a. Graph the equation. What are the intercepts? b. Write the factored form of the equation. c. How are the intercepts related to the original equation? d. Write each equation in factored form. Verify each answer by graphing. i. y 2 49 ii. y 2 2 iii. y 47 iv. y 2 28 e. An epression in the form a2 b2 is called a difference of two squares. Based on your work in 2a d, make a conjecture about the factored form of a2 b2. f. Graph the equation y 2 4. How many intercepts can you see? g. Eplain the difficulty in trying to write the equation in 2f in factored form. The roots of 0 2 are 4 and 4 3. Kayleigh says that the roots of 0 2 are because (4)2 and 4 and 4 because (4)2 and (4)2. (4)2. Derek tells Kayleigh that there are no roots for There are no roots this equation. Who is correct and why? for this equation. 4. Reduce the rational epressions by dividing out common factors from the numerator and denominator. State any restrictions on the variable. ( 2)( 2) a b. a. ( 2)( 3) ( 4)( 2) a c. d e. 2 9 Review. Multiply and combine like terms. a. ( 2)( 2) b. (3 )( 4) c. 2(2 3)( 2). Edward is responsible for keeping the stockroom packed with the bestselling merchandise at the Super Store. He has collected data on sales of the new video game MathaMagic. Week Games sold [Data sets: GMWK, GMSLD] a. Find a quadratic model in verte form that fits the data. Let w represent the week number and let s represent the number of games sold. b. If the pattern continues, in what week will people stop buying the game? c. How many total games will have been sold when people stop buying the game? d. There are 000 games left in the stockroom at the start of week. How many more should Edward buy? LESSON 9.4 Factored Form 2
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