Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic functions. There are practice questions that reflect the types of assessment questions that may be used on semester eams. There is also a list of reflection questions that should be covered to help students demonstrate practice standards. It is suggested that teachers work these problems before instruction to identify areas where students may have difficulty, then include appropriate instruction and review as necessary. Math Background Previously, you Identified and graphed linear functions Transformed linear functions Solved linear equations Factored quadratic polynomials, including perfect square trinomials In this unit you will study Identifying and graphing quadratic functions Transforming quadratic equations Solving quadratic equations Using factoring to graph quadratic functions and solve quadratic equations Unit skills: Solve quadratic equations by inspection, using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. You can use the skills in this unit to Determine the maimum height of a ball thrown into the air Graph higher degree polynomials in future math courses Solve problems about free fall such as height of launched objects Overall Big Ideas Quadratic equations can be written in various forms. There are various methods we can use to solve quadratic equations. The quadratic formula can be derived. Essential Questions What makes an equation a quadratic equation? How can algebra be used to solve quadratic equations? How many solutions can quadratic equations have? Are all solutions always real numbers? How can I decide which method to use to solve quadratic equations? Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 1 of 40 5/0/013
Concepts: Need to know about quadratic equations in one variable Vocabulary to know Forms of quadratic equations p; 5 b c; 4 5 a b c 0; 8 6 0 ( ) ; ( 3) 5 ( h) 4p p q Methods of solution Inspection Square roots Complete the square Graph Quadratic formula Factor Types of solutions (quadratic formula) Real Comple Skills: Be Able to Do Solve quadratic equations in one variable Use the method - completing the square Derive the quadratic formula Solve by inspection ( 5 ) This is the simplest form of taking square roots. Solve by taking square roots Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page of 40 5/0/013
Solve by completing the square Future uses, conics, trig functions, zeros, transformations, etc. Solve by using quadratic formula How to derive it. Solve by factoring Use lots of hints, patterns, and effective practice. Solve by graphing Write comple solutions as a bi (using quadratic formula) Applications for specific parts of the graph. Target: only as it applies to QF at this point. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 3 of 40 5/0/013
Skill: Solve quadratic equations using square roots. N.RN.A. Etend the properties of eponents to rational eponents. Rewrite epressions involving radicals and rational eponents using the properties of eponents. N.RN.B.3 Use properties of rational and irrational numbers. Eplain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. A.REI.B.4b Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate. Square Root: REVIEW: Square Roots Radical Sign: Radicand: the number beneath the radical sign Positive (Principal) Square Root: 9 3 Negative Square Root: 9 3 Review: Simplifying Square Roots A square root is simplified if the radicand has no perfect square factor (other than 1) and there is no radical in the denominator of a fraction. E 1: Simplify the square root 7. Method 1 Step One: Find the largest perfect square that is a factor of 7. 36 Step Two: Rewrite 7 as a product using 36 as a factor. 36 Step Three: Rewrite as the product of two radicals. 36 Step Four: Evaluate the square root of the perfect square. 6 Method Step One: Rewrite 7 as a product of prime factors. 3 3 33 Step Two: Find the square root of each pair of factors. E : Simplify the epression 3 8. 3 3 3 6 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 4 of 40 5/0/013
8 We must rationalize the denominator by multiplying by 1. 8 4 4 6 8 8 Now simplify the radical and the fraction. 6 4 6 6 8 8 3 8 8 8 4 8 Quadratic Equation: an equation that can be written in the standard form a b c 0, a 0 Solving a Quadratic Equation by Finding Square Roots: to use this method, the quadratic equation must be able to be written in the form 0, 0 a c b or a c E 3: Solve the equation 16. Step One: Find the square root of both sides. 4 16 Step Two: Solve for. (Note that there are two solutions.) 4, 4 This can also be written 4 E 4: Solve the equation 4 4. Step One: Isolate the squared epression. 44 0 Step Two: Find the square root of both sides. 0 0 Step Three: Solve for. (Note that there is one solution.) 0 E 5: Solve the equation 3. Step One: Isolate the squared epression. 3 1 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 5 of 40 5/0/013
Step Two: Find the square root of both sides. 1 1 1 is not a real number Step Three: Solve for. (Note that there is no solution.) This equation has no real solution. E 6: Solve the equation 3 108 0. Step One: Isolate the squared epression. 3 108 36 Step Two: Find the square root of both sides. 6 36 Step Three: Solve for the variable. 6 or 6 E 7: Solve the equation 7 43. Step One: Isolate the squared epression. Step Two: Find the square root of both sides. 7 43 50 50 5 Step Three: Solve for the variable. 5 or 5 E 8: Solve the equation n 5 16. Step One: Isolate the squared epression. n 5 81 Step Two: Find the square root of both sides. n 5 81 n 5 9 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 6 of 40 5/0/013
Step Three: Solve for the variable. n 5 9 n 5 9 n 14 n 4 n 7 n E: Solve the equation 1 8 7 4 a, Step One: Isolate the squared epression. a 8 8 Step Two: Find the square root of both sides. Step Three: Solve for the variable. a 8 8 a 8 7 a 8 7 a 8 7 a 8 7 a 8 7 a 8 7 Note: The ( plus or minus ) symbol is used to write both solutions in a shorter way. In set notation, the solutions would be written 8 7,8 7. Real-Life Application: Free Fall On Earth, the equation for the height (h) of an object for t seconds after it is dropped can be modeled by the function h 16t h, where h0 is the initial height of the object. 0 E 9: A ball is dropped from a height of 81 ft. How long will it take for the ball to hit the ground? Use the free-fall function. h0 81, h 0 h 16t h0 Initial height is 81 ft. The ball will hit the ground when its height is 0 ft. Solve for t. 0 16t 81 16t 81 81 t 16 9 t 4 9 9 t, 4 4 Solution: Since time is positive, the only feasible answer is 9 4.5 seconds You Try: Solve the equation 7 10 1. QOD: Why do some quadratic equations have two, one, or no real solution? Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 7 of 40 5/0/013
Sample Practice Question(s): 1. Solve the equation 4 5 0. A. B. C. D. 5 5 4 5 5, 5 5, 4 4. Solve. a. b. Sample Nevada High School Proficiency Eam Questions (taken from 009 released version H): An equation is shown below. 5 15 What is the solution set of the equation? A 5 B 5,5 C 5 D 1.5,1.5 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 8 of 40 5/0/013
Characteristics of Quadratic Functions and Graphs Skill: Compare characteristics of a given family of quadratic functions. Skill: Determine the domain and range of quadratic equations algebraically and graphically. F.IF.B.4 Interpret functions that arise in applications in terms of the contet. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maimums and minimums; symmetries; end behavior; and periodicity. F.IF. B.5 Interpret functions that arise in applications in terms of the contet. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes F.IF.C.7a Analyze functions using different representations. Graph functions epressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maima, and minima. F.IF.C.8a Analyze functions using different representations. Write a function defined by an epression in different but equivalent forms to reveal and eplain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, etreme values, and symmetry of the graph, and interpret these in terms of a contet. F.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Note to teachers: The complete notes for the characteristics Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 9 of 40 5/0/013
Notes Quadratic Function: a function that can be written in the form Parabola: the U-shaped graph of a quadratic function y a b c, when a 0 Verte: the highest or lowest point on a quadratic function (maimum or minimum) Ais of Symmetry: the vertical line that passes through the verte of a quadratic function Verte Ais of Symmetry Verte Ais of Symmetry Domain and Range: Domain of a Quadratic Function: all real numbers Range of a Quadratic Function: If the parabola opens up, then the range is all values of y greater than or equal to the y- coordinate of the verte. If the parabola opens down, then the range is all values of y less than or equal to the y- coordinate of the verte. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 10 of 40 5/0/013
Graphing a Parabola E 10: Graph the quadratic function Step One: Make a table of values (t-chart). 3 1 0 1 3 y 9 4 1 0 1 4 9 y. Step Two: Plot the points on a coordinate grid and connect to draw the parabola. Note: The verte is 0,0, and the ais of symmetry is 0. E 11: Graph the parabola y Step One: Make a table of values (t-chart). 3 1 0 1 3 y 9 4 1 0 1 4 9 Step Two: Plot the points on a coordinate grid and connect to draw the parabola. Note: The verte is 0,0, and the ais of symmetry is 0. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 11 of 40 5/0/013
Comparing y and y : The verte is 0,0, and the ais of symmetry is 0 for both graphs. When a is positive, the parabola opens up; when a is negative, the parabola opens down. Activity: Transformations with y and y. Use the graphing calculator to graph the quadratic functions. Describe the effect on the graphs of y and dotted line.) y. (Note: In the calculator graphs shown, y or y is graphed as a 1. y Compare to y Verte: Same 0,0 Opens narrower than y. y 1 3 Compare to y Verte: Same 0,0 Opens wider than y 3. y 3 Compare to y Verte: Up 30,3 Opens the same as y 4. y 4 Compare to y Verte: Down 40, 4 Opens the same as y Conclusions (sample): For quadratic functions of the form y a c a 1 Opens down Narrower than y 1 a 0 Opens down Wider than 0a 1 Opens up Wider than a 1 Opens up Narrower than y y y c 0 Verte moves down c units c 0 Verte moves up c units Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 1 of 40 5/0/013
Standard Form of a Quadratic Function: y a b c Verte: the -coordinate of the verte is b Ais of Symmetry: a b a Graphing a Quadratic Function in Standard Form E 1: Graph the quadratic function and range. y 6 1. State the verte, ais of symmetry, domain, Step One: Find the -coordinate of the verte. a1, b 6 6 3 1 Step Two: Make a table of values. When choosing -values, use the verte, a few values to the left of the verte, and a few values to the right of the verte. 0 1 3 4 5 6 y 1 6 9 10 9 6 1 y-coordinate of verte: y 3 6 3 1 9 18 1 10 Note: When calculating the y-coordinate of points to the right and left of the verte, notice the symmetry. Step Three: Plot the points from the table and draw the parabola. Verte: 3, 10 Ais of Symmetry: 3 Domain: all real numbers Range: y 10 E 13: Graph the quadratic function 1 1. State the verte and ais of symmetry. y 3 Step One: Find the -coordinate of the verte. 1 a, b 3 3 3 3 1 1 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 13 of 40 5/0/013
Step Two: Make a table of values. When choosing -values, use the verte, a few values to the left of the verte, and a few values to the right of the verte. (Note: Because of the fraction, you may want to choose values that will guarantee whole numbers for the y-coordinates.) 9 7 5 3 1 1 3 y 14 4 4 4 14 Step Three: Plot the points from the table and draw the parabola. Verte: 3,4 Ais of Symmetry: 3 E 14: Graph the quadratic function y 8 10. Step One: Find the -coordinate of the verte. a, b 8 8 8 4 Step Two: Make a table of values. When choosing -values, use the verte, a few values to the left of the verte, and a few values to the right of the verte. 1 0 1 3 4 5 y 0 10 16 18 16 10 0 y-coordinate of verte: y 8 10 8 16 10 18 Note: When calculating the y-coordinate of points to the right and left of the verte, notice the symmetry. Step Three: Plot the points from the table and draw the parabola. Verte:, 18 Ais of Symmetry: Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 14 of 40 5/0/013
Using a Quadratic Model E 15: A basketball s path can be modeled by y 16 15 6, where represents time (in seconds) and y represents the height of the basketball (in feet). What is the maimum height that the basketball reaches? Graph the function and find the maimum (in the Calc menu). The maimum is the verte. The maimum height of the basketball is the y-coordinate of the verte, which is approimately 9.5ft. You Try: Find the verte and ais of symmetry for the following quadratic function. Determine if the parabola will open up or down. Then graph the parabola. y QOD: How many points does it take to determine a unique parabola? Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 15 of 40 5/0/013
Sample Practice Question(s): 1. Which of the following is the graph of y 6? Why? A. B. C. D. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 16 of 40 5/0/013
. Which equation best represents the following graph? Why? A. B. C. D. y 8 y 8 y 8 y 8 3. Which of the following are true statements about the graph of y 8 4? I. Opens Up II. Opens Down III. Ais of symmetry = 4 IV. Ais of symmetry = 4 B. I and III only C. I and IV only D. II and III only E. II and IV only Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 17 of 40 5/0/013
4. Find the verte of the parabola: A. 6, 7 y B. 3, 11 C. 3, 61 D. 6, 151 1 7 5. What is the domain and range of the function below? y 4 shown in the graph A. Domain: all real numbers Range: all real numbers B. Domain: Range: y 4.5 C. Domain: all real numbers Range: y 4.5 D. Domain: Range: all real numbers Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 18 of 40 5/0/013
Sample Nevada High School Proficiency Eam Questions (taken from 009 released version H): A painter is designing a mural. The mural will be shaped like a rectangle. The length of the mural will be 3 times the width of the mural. Which graph shows the relationship between the width of the mural () and the area of the mural (y)? Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 19 of 40 5/0/013
Skill: solve quadratic equations using graphic techniques. Skill: graph quadratic equations and find possible solutions to those equations using coordinate geometry. F.IF.C.7 Analyze functions using different representations. Graph functions epressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. F.IF.B.4 Interpret functions that arise in applications in terms of the contet. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maimums and minimums; symmetries; end behavior; and periodicity. Solving a Quadratic Equation by Graphing Step One: Write the equation in the form a b c 0. Step Two: Graph the function y a b c. Step Three: Find the zero(s) or root(s) of the function. These are the solution(s) to the equation. Note: The words -intercept, zero, root, and solution can be used interchangeably for the above value. E: Solve the equation 1 8 by graphing. Then check by solving algebraically. Step One: Write the equation in the form a b c 0. 1 8 0 Step Two: Graph the function y a b c. y 1 8 Verte: b 0 0 a 1 1 y 0 8 8 0, 8 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 0 of 40 5/0/013
Step Three: Find the -intercept(s) of the function. The zeros are at 4 and 4, so the solutions are 4,4. Solve algebraically: 1 8 16 4 4 16 E 16: Solve the equation 3 by graphing. Step One: Write the equation in the form 3 0 a b c 0. Step Two: Graph the function y a b c. y 3 Verte: b 3 3 a 1 3 3 9 9 9 18 8 1 y 3 4 4 4 4 4 3 1, 4 Step Three: Find the -intercept(s) of the function. The zeros are at 1 and, so the solutions are 1,. E 17: Solve the equation 4 4 by graphing. Step One: Write the equation in the form a b c 0. 4 4 0 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 1 of 40 5/0/013
Step Two: Graph the function y 4 4 y a b c. Verte: b 4 4 a 1 y 4 4 4 8 4 0,0 Step Three: Find the -intercept(s) of the function. The root is at, so the solution is. Check: 4 4 4 4 4 4 8 E 18: Solve the equation 3 0 graphically. Step One: Write the equation in the form a b c 0. 3 0 Step Two: Graph the function y 3 y a b c. Verte: b 0 0 0 a 3 6 y 0 3 3 0,3 Step Three: Find the -intercept(s) of the function. There is no zero, so this equation has no real solution. Using a Graphing Calculator to Solve Quadratic Equations E 19: Approimate the solution(s) of 1 4 using a graphing calculator. Step One: Write the equation in the form a b c 0. 41 0 Step Two: Graph the function y a b c. y 4 1 Step Three: Find the zero(s) of the function. 4.36,0.36 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page of 40 5/0/013
You Try: Solve the quadratic equation 6 3 graphically. Then check your answer algebraically. QOD: How can you tell from the graph of a quadratic function if the equation has one, two, or no solution? Sample Practice Question(s): The graph of y 1 has how many -intercepts? A. 1 B. C. 1 D. 0 Sample Nevada High School Proficiency Eam Questions (taken from 009 released version H): Fiona is designing a skateboard park. One skating area in the park will be shaped like the parabola that is described by the equation below. 1 18 45 6 y A sketch of Fiona s design for the skating area is shown below. What is the distance across the top of the skating area? A 1 units B 14 units C 15 units D 18 units Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 3 of 40 5/0/013
Skill: solve quadratic equations by factoring. A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. For eample, see 4 y 4 as ( ) (y ), thus recognizing it as a difference of squares that can be factored as ( y )( + y ). A.SSE.B.3a Write epressions in equivalent forms to solve problems. Choose and produce an equivalent form of an epression to reveal and eplain properties of the quantity represented by the epression. Factor a quadratic epression to reveal the zeros of the function it defines. A.REI.B.4b Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. Review: Factoring Quadratic Trinomials into Two Binomials (Using the ac method or splitting the middle term.) NOTE: You may want to see the Unit 07 Factor by Splitting the Middle Term notes AND Unit 07 Factoring Quadratic Trinomial notes on www.rpdp.net. Factoring a b c a, 1 E 0: Factor 7 1. Find two integers such that their product is 1 and their sum is 7. 4 and 3 Write the two binomials as a product. 4 3 Factoring a b c a, 1 E 1: Factor 7 3. Step One: Multiply a c. 3 6 Step Two: Find two integers such that their product is ac6 and their sum is b 7. 6 and 1 Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two. 6 1 3 Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 4 of 40 5/0/013
6 1 3 3 1 3 Step Five: If Step Four was done correctly, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 1 3 E : Factor 5 7. Step One: Multiply a c. 5 10 Step Two: Find two integers such that their product is ac10 and their sum is b 7. and 5 Step Three: Rewrite ( split ) the middle term as a sum of two terms using the numbers from Step Two. 5 5 Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair. 5 5 5 15 Step Five: If Step Four was done correctly, there should be a common binomial factor. Factor this binomial out and write what remains from each term as the second binomial factor. 5 1 Recall: Special Factoring Patterns Difference of Two Squares: a b a ba b Perfect Square Trinomial: a ab b a b a ab b a b Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 5 of 40 5/0/013
Zero Product Property: If the product of two factors is 0, then one or both of the factors must equal 0. E 3: Solve the equation 3 1 0 using the zero product property. Since one or both of the factors must equal 0, we will solve the two equations 3 0 and 1 0. 30 3 10 1 Solutions: 1, 3 Solving a Quadratic Equation by Factoring E 4: Solve the equation 5 6 by factoring. Step One: Write the equation in standard form. 5 6 0 Step Two: Factor the quadratic. 3 0 Step Three: Set each factor equal to zero and solve. Note: Check this answer by graphing on the calculator. 3 0 0 3, 3 E 5: Solve the equation 4 8. Step One: Write the equation in standard form. Step Two: Factor the quadratic using the ac method. 3 58 0 a c 4 b 5 8 and 3 3 8 3 8 0 3 8 1 0 3 8 1 3 8 0 Step Three: Set each factor equal to zero and solve. The solutions can be written in set notation: 3 8 0 1 0 8 3 1 8 1, 3 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 6 of 40 5/0/013
E 6: Solve the equation 5 30 9 y y. Step One: Write the equation in standard form. 9y 30y 5 0 Step Two: Factor the quadratic. 3y 30y5 y Note: 5 3 30y 3y 5 0 Step Three: Set each factor equal to zero and solve. The solution can be written in set notation: 3y 5 0 y 5 3 5 3 Zero(s) of Quadratic Functions: the -value(s) where the function intersects the -ais To find the zero(s), factor the quadratic and set each factor equal to 0. Note: We can graph quadratic functions by plotting the zeros. The verte is halfway between the zeros. E 7: Find the zero(s) of the quadratic function y 3and graph the parabola. Step One: Factor the quadratic polynomial. Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. y 3 3 1 y 3 0 1 0 3 1 3 1 1 y 1 1 3 1 3 4 Step Four: Plot the points and sketch the parabola. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 7 of 40 5/0/013
You Try: Solve the quadratic equation 5t 5 4t 6 by factoring. QOD: What must be true about a quadratic equation before you can solve it using the zero product property? Sample Practice Question(s): 1. What is the solution set for the following equation? 10 9 0 A. { 9, 1} B. { 9, 1} C. { 1, 9} D. {1, 9}. Which of the following equations has roots of 7 and 4? A. 7 4 0 B. 7 4 0 C. 7 4 0 D. 7 4 0 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 8 of 40 5/0/013
Skill: solve quadratic equations by completing the square. A.SSE.A. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. A.SSE.B.3b Write epressions in equivalent forms to solve problems. Complete the square in a quadratic epression to reveal the maimum or minimum value of the function it defines. A.REI.B.4a Solve equations and inequalities in one variable. Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( p) = q that has the same solutions. Derive the quadratic formula from this form. Review: Factoring a Perfect Square Trinomial a ab b a b a ab b a b NOTE: You may want to review the Unit 08 Completing the Square Notes on www.rpdp.net. Completing the Square: writing an epression of the form to factor it as a binomial squared To complete the square of b, we must add b as a perfect square trinomial in order b. Teacher Note: Algebra Tiles work well to illustrate completing the square. See Page 79 for an activity. E 8: Find the value of c such that 10 c is a perfect square trinomial. b 10, therefore we must add 10 5 5 Note: 10 5 c to complete the square. Solving a Quadratic Equation by Completing the Square E 9: Solve 8 7 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides). Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 9 of 40 5/0/013 8 7 0 8 7 8 8 8 7 816 9 Step Four: Factor the perfect square trinomial. 4 9
Step Five: Take the square roots of both sides. 4 3 4 9 4 3 4 3 Step Si: Solve for the variable. 1 7 The solution set is 7, 1. Check your answer by factoring. E 30: Solve 1 4 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add to both sides). 1 4 0 6 0 6 6 6 6 6 9 11 Step Four: Factor the perfect square trinomial. 3 11 Step Five: Take the square roots of both sides. Step Si: Solve for the variable. The solution set is 3 11, 3 11 3 11 3 11 3 11 3 11 3 11 3 11 E 31: Solve 3 0 by completing the square. Step One: Rewrite to make the lead coefficient 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add Step Four: Factor the perfect square trinomial. to both sides). 3 0 3 1 1 3 1 13 4 4 1 13 4 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 30 of 40 5/0/013
Step Five: Take the square roots of both sides. Step Si: Solve for the variable. 1 13 The solutions are. 1 13 4 1 13 1 13 1 13 1 13 1 13 You Try: Solve by completing the square. 4 011 0 QOD: Describe why adding b to b makes it a perfect square trinomial. Sample CCSD Common Eam Practice Question(s): What are the roots (solutions) of A. 1 3, 1 3 B. 1 5, 1 5 1 0? C. D. 1 5 1 5, 1 5 1 5, Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 31 of 40 5/0/013
Skill: solve quadratic equations using the quadratic formula. A.SSEA.. Interpret the structure of epressions. Use the structure of an epression to identify ways to rewrite it. A.REI.B.4a Solve equations and inequalities in one variable. Use the method of completing the square to transform any quadratic equation in into an equation of the form ( p) = q that has the same solutions. Derive the quadratic formula from this form. A.REI.B.4b Solve equations and inequalities in one variable. Solve quadratic equations by inspection (e.g., for = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives comple solutions and write them as a ± bi for real numbers a and b. A.REI.A.1 Understand solving equations as a process of reasoning and eplain the reasoning. Eplain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Notes: Deriving the Quadratic Formula by Completing the Square Solve the quadratic equation a b c 0 by completing the square. Step One: Rewrite so that the lead coefficient is 1. Step Two: Take the constant term to the other side. b Step Three: Complete the square (add Step Four: Factor the perfect square trinomial. Step Five: Take the square roots of both sides. to both sides). a b c 0 a a a a b c 0 a a b c a a b b c b a a a a b b 4ac b a 4a 4a b b 4ac a 4a b b 4ac a 4a b b 4ac a 4a Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 3 of 40 5/0/013
Step Si: Solve for the variable. b b 4ac b b 4ac a a a a a a b b 4ac b b 4ac The Quadratic Formula: To solve a quadratic equation in the form b b 4ac. a a b c 0, use the formula Note: To help memorize the quadratic formula, sing it to the tune of the song Pop Goes the Weasel. E 3: Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 81 0 Step Two: Identify a, b, and c. a 1, b 8, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a 8 8 411 1 8 64 4 8 60 8 15 Step Four: Simplify. 4 15 The solution set is 4 15,4 15 E 33: Solve the quadratic equation 5 1 6 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 6 51 0 Step Two: Identify a, b, and c. a 6, b 5, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a 5 5 46 1 6 Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 33 of 40 5/0/013
Step Four: Simplify. The solution set is 1 1, 6 5 5 4 5 49 5 7 1 1 1 5 7 1 5 7 1 1 1 1 1 1 6 E 34: Solve the quadratic equation 1 4 9 0 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 9 1 4 0 Step Two: Identify a, b, and c. a 9, b 1, c 4 Step Three: Substitute the values into the quadratic formula. Step Four: Simplify. The solution set is 3. b b 4ac a 1 1 494 9 1 144 144 1 0 1 18 18 18 3 E 35: Solve the quadratic equation 3 0 using the quadratic formula. Step One: Rewrite in standard form (if necessary). 3 0 Step Two: Identify a, b, and c. a, b, c 3 Step Three: Substitute the values into the quadratic formula. b b 4ac a 43 4 4 0 Step Four: Simplify. 4 4 There is no real solution to the quadratic equation because 0 is not a real number. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 34 of 40 5/0/013
You Try: Solve the equation 6 3 using the quadratic formula. QOD: Write a conjecture about how the radicand in the quadratic formula relates to the number of solutions that a quadratic equation has. Sample Practice Question(s): 1. What are the roots (solutions) of A. 1 3, 1 3 B. 1 5, 1 5 1 0? C. D. 1 5 1 5, 1 5 1 5,. Which of the following is the correct use of the quadratic formula to find the solutions of the equation 7 5? A. B. C. D. 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, 7 7 45 7 7 4 5, Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 35 of 40 5/0/013
Discriminant: the discriminant of the quadratic equation a b c 0 is b 4ac Note: The discriminant is the radicand of the quadratic formula! Determining the Number of Real Solutions of a Quadratic Equation Using the Discriminant Teacher Note: Students should have come up with this in the QOD. If If If b b b 4ac 0, then there are no real solutions. 4ac 0, then there is one solution. 4ac 0, then there are two real solutions. E 36: Determine the number of real solutions that the equations have. 1. 3 1 Rewrite the equation in standard form. 3 1 0 a 3, b 1, c 1 Find the discriminant. b ac 4 1 4 3 1 11 11 Determine the number of real solution(s). b 4ac 11 0, so there are no real solutions.. 45 Rewrite the equation in standard form. 5 4 0 a 1, b 5, c 4 Find the discriminant. b ac 4 5 4 1 4 5 16 41 Determine the number of real solution(s). b 4ac 41 0, so there are two real solutions. 3. 9 1 4 0 Rewrite the equation in standard form. 9 1 4 0 a 9, b 1, c 4 Find the discriminant. b ac 4 1 4 9 4 144 144 0 Determine the number of real solution(s). b 4ac 0, so there is one real solution. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 36 of 40 5/0/013
Determining the Number of -Intercepts of a Quadratic Function Using the Discriminant Because the -intercepts of y a b c are the same as the zeros of the equation a b c 0, we can use the discriminant to determine the number of -intercepts that a quadratic function has. E 37: Sketch the graph of a quadratic function with a negative discriminant. Because the discriminant, b 4ac 0, the function will have no -intercept. A sample answer is shown in the graph. Note: Any parabola which does not intersect the -ais is an acceptable answer. Application Problem E 38: A baton twirler tosses a baton into the air. The baton leaves the twirler s hand 6 feet above the ground and has an initial vertical velocity of 45 feet per second. This can be modeled by the equation h 16t 45t 6, where h is the height (in feet) and t is the time (in seconds). The twirler wants her baton to reach at least 40 feet. Will the baton reach that height? Substitute h 1. Write in standard form. 40 16t 45t 6 0 16t 45t 34 a 16, b 45, c 34 Find the discriminant. b ac 4 45 4 16 34 05 176 151 Since the discriminant is less than 0, this equation has no real solution. Therefore, the baton could not reach 40 feet. How high will the baton reach? Graph the function h t t 16 45 6. Find the maimum (verte). The baton will reach approimately 37.64 ft. You Try: Find values for c so that the equation will have no real solution, one real solution, and two real solutions. c 3 0 QOD: Write a quadratic equation which can be factored. Find its discriminant. Teacher Note: Have students share their answers to the QOD and allow students to make a conjecture for how to determine if a quadratic polynomial is factorable using the discriminant. (It must be a perfect square.) Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 37 of 40 5/0/013
Sample Practice Question(s): The graph of y 1 has how many -intercepts? A. 1 B. C. 1 D. 0 Transformations of quadratic functions 1. How would the graph of the function be affected if the function were changed to? a. The graph would shift 5 units to the left. b. The graph would shift 5 units down. c. The graph would shift 5 units up. d. The graph would shift 3 units down.. How would the graph of the function be affected if the function were changed to? a. The graph would shift units up. b. The graph would shift 5 units up. c. The graph would shift units to the right. d. The graph would shift 5 unit down. 3. How would you translate the graph of to produce the graph of a. translate the graph of down 4 units b. translate the graph of up 4 units c. translate the graph of left 4 units d. translate the graph of right 4 units 4. Compare the graph of with the graph of. a. The graph of g() is wider. b. The graph of g() is narrower. c. The graph of g() is translated 6 units down from the graph of f(). d. The graph of g() is translated 6 units up from the graph of f(). 5. Compared to the graph of, the graph of is. a. narrower and translated down c. wider and translated down b. narrower and translated up d. wider and translated up Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 38 of 40 5/0/013
6. Four bowls with the same height are constructed using quadratic equations as their shapes. Which bowl has the narrowest opening? a. c. Bowl Bowl b. Bowl d. Bowl 7. What is the parent function for? a. c. b. d. 8. Use the quadratic function f() = ( + 3)( 1), to graph the function. a. y c. 10 8 6 4 10 8 6 4 y 10 8 6 4 4 6 8 10 4 6 8 10 10 8 6 4 4 6 8 10 4 6 8 10 b. 10 y d. 10 y 8 8 6 6 4 4 10 8 6 4 4 6 8 10 4 6 8 10 10 8 6 4 4 6 8 10 4 6 8 10 9. Use the quadratic formula to solve the equation Describe how you can use the graph of the equation to verify your solutions of the equation Sketch this graph and verify your solutions. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 39 of 40 5/0/013
10. Eplain the zero-product property and why it is useful. Give an eample to illustrate your eplanation. Sample answer: The zero-product property states that if a product of factors is equal to zero, then one or more of the factors is equal to zero. For eample, in the equation the polynomial can be factored as ( + 7)( ). Since this factored polynomial is equal to zero, then the value of at least one of the factors is 0. This fact is useful because you can easily solve the equations and to find the solutions = 7 and =. Construct viable arguments for the following reflection questions. 1. Describe how to use the discriminant to find the number of real solutions to a quadratic equation.. Choose a method to solve 5 4 0 and eplain why you chose that method. 3. Describe how the discriminant can be used to determine if an object will reach a given height. 4. Eplain why there are no solutions to the quadratic equation 5. Describe how to estimate the solutions of 4 16 6. How do you find the zeros of a function from its graph? 9.. What are the approimate solutions? 7. Describe how to find the ais of symmetry of a quadratic function if its graph crosses the ais. Describe how to find the ais of symmetry of a quadratic function if its graph does not cross the ais? 8. Eplain how to graph a quadratic function. 9. What do the verte and zeros represent in the following situation. As Joe dives into his pool, his height in feet above the water can be modeled by the function f ( ) 16 16 1 where is the time in seconds after he begins diving. Find the maimum height of the dive and the time it takes Joe to reach this height. Then find how long it takes him to reach the water. 10. Eplain two ways to solve 6 0. How are these two methods similar? 11. Describe the relationships among the solutions, the zeros, and the -intercepts of y 4 1. Alg I Unit 08 Notes QuadFuncEqRadicalsrev Page 40 of 40 5/0/013