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1 Big Idea: A quadratic equation in the form a b c 0 has a related function f ( ) a b c. The zeros of the function are the -intercepts of its graph. These -values are the solutions or roots of the related quadratic equation. A quadratic equation can have one real solution, two real solutions, or no real solutions. Objectives: (Common Core) 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. 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. 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. Skills: Using Intercept form to graph Solve quadratic equations by graphing Write quadratic equations in standard form using intercepts. Write quadratic equations given 3 points. Vocabulary: -intercept: the -coordinate of the point where the curve intersects the -ais Intercept Form of a Quadratic Function: y a p q -Intercepts: pq, -Coordinate of Verte: p q (verte is halfway between the -intercepts) Models: A quadratic equation can have one real solution, two real solutions, or no real solutions. Two real solutions One Real Solution No real solution Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 1 of 44 6/14/013

2 Graphing a Quadratic Function in Intercept Form E 1: Graph the quadratic function y 5 1. Step One: Identify the -intercepts. -intercepts are 5 and 1 Note: It may be helpful to write the equation as y 5 1 Step Two: Identify the verte. y-coordinate of verte: 51 -coordinate of verte: y Step Three: Plot the -intercepts and the verte and draw the parabola. Writing the Equation of a Quadratic Function in Standard Form E : Write and equation for the parabola in standard form. The -intercepts (zeros) of the parabola are at and 3. So the intercept form of the quadratic equation is y a 3. To solve for a, we will choose a point on the parabola and substitute it into the equation for y,. Choose,. a 3 4a 1 a 1. So the intercept form of the equation is y 3 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page of 44 6/14/013

3 To rewrite in standard form, multiply the binomials and distribute the constant. 1 1 y y Writing the Equation of a Quadratic Function Given Three Points E 3: Write a quadratic function in standard form for the parabola whose graph passes through the points,, 3,4, and 0,. Use the standard form. Substitute each point in for, y a b c system of three equations for a, b, and c. y and solve the remaining a b c 4a b c 4 a 3 b 3 c 4 9a 3b c a 0 b 0 c c Since c, we can substitute this value into the first two equations. 4a b 0 4ab Solve the system of the remaining two equations. 4 9a 3b 6 9a3b We will use the substitution method. 0 4ab 4a b a b 6 9a 3 a 6 3a a 0 4 b 8b 4 b Substitute the values for a, b, and c into the standard form equation y a b c. y 4 You Try: 1. Write the verte form, intercept form, and standard form of the parabola shown in the graph.. Write the equation of the quadratic function that passes through the points, 1, 1,11, and,7. QOD: How can you tell from the graph of a quadratic function if the equation has one, two, or no solution? Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 3 of 44 6/14/013

4 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 4 of 44 6/4/013

5 Sample Practice Question(s): The graph of y 1 has how many -intercepts? A. 1 B. C. 1 D. 0 Using Standard Form, Verte Form, and Intercept Form: You Try: Find the verte and ais of symmetry for the following quadratic functions. Determine if the parabola will open up or down. Then graph the parabola. y y 4 1 y 3 9 Closure: Describe the three forms of an equation of a quadratic function. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 5 of 44 6/4/013

6 Sample Questions: 1. f() = ( + 3)( 1), graph the function. a y c y b y d y ANS: B. Solve the equation by graphing. 10 y Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 6 of 44 6/4/013

7 10 y ANS: = 1 and = 4 3. Solve the equation by graphing, approimate your answers to the nearest tenth. 10 y y ANS: , 1 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 7 of 44 6/4/013

8 Big Idea: To solve a quadratic equation by factoring, make sure the equation is in the form a b c 0, factor the polynomial epression, set each factor equal to zero, and then solve the resulting equations. Techniques used to solve quadratic equations include the techniques for factoring general trinomials, perfect square trinomials, and difference of squares. Objectives: (Common Core) 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. 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. 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. A.SSEA.. 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 Skills: Solve quadratic equations by factoring. Determine a quadratic function from its roots. Vocabulary: Standard Form of a Quadratic Equation: a b c 0 Zero Product Property: If the product of two factors is 0, then one or both of the factors must equal 0. Roots: The solutions to a quadratic equation of the form a b c 0 are roots. The roots of an equation are the values of the variable that make the equations true. 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. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 8 of 44 6/4/013

9 Review: Factoring Quadratic Trinomials into Two Binomials (Using the ac method or splitting the middle term.) Factoring a b c a, 1 E 1: 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 : 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 (order does not matter when splitting the middle term) Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair 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 3: Factor 5 7. Step One: Multiply a c 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 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 9 of 44 6/4/013

10 Step Four: Factor by grouping. Group the first terms and last terms and factor out the GCF from each pair 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 Special Factoring Patterns: Memorize these! Difference of Two Squares: a b a ba b Perfect Square Trinomial: a ab b a b a ab b a b E 4: Factor y. This appears to be a difference of two squares, since each term is a perfect square. Rewrite each term as a monomial squared then use the pattern to factor. 3 4y 3 4y 3 4y E 5: Factor 49m 14mn n. This appears to be a perfect square trinomial. Rewrite the first and last terms as a monomial squared and check to see if the middle term is twice the product of these monomials. Then use the pattern to factor. 7m 14mn n 7mn 7m n 14mn ) (Check: Factoring a GCF Monomial E 6: Factor 7 50y completely. Step One: Factor out the GCF of. 36 5y Step Two: Factor the remaining polynomial. 6 5y6 5y Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 10 of 44 6/4/013

11 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 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 3and graph the parabola. y y y Step Four: Plot the points and sketch the parabola. Solving a Quadratic Equation by Factoring E 8: Solve the equation 4 8. Step One: Write the equation in standard form. Step Two: Factor the quadratic using the ac method a c 4 b 5 8 and Step Three: Set each factor equal to zero and solve. The solutions can be written in set notation: , 3 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 11 of 44 6/4/013

12 E 9: Solve the equation y y. Step One: Write the equation in standard form. 9y 30y 5 0 Step Two: Factor the quadratic. 3y 30y5 y Note: y 3y 5 0 Step Three: Set each factor equal to zero and solve. 3y 5 0 y 5 3 The solution can be written in set notation: 5 3 Using Zeros to Write Functions Rules E10: Write a quadratic function in standard form with zeros and 1. or 1 Write the zeros as solutions for two equations. 0 or 1 0 Rewrite each equation so that it equals 0. ( )( 1) 0 Apply the converse of the Zero Product Property to write a product that equals 0. 0 Multiply the binomials. f ( ) replace 0 with f( ) Check: Graph the function f ( ) on a calculator. The graph shows the original zeros of and 1. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 1 of 44 6/4/013

13 Application: Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile on Earth after t seconds is given below. h( t) 16t vot ho Constant due to Earth s gravity in ft / sec Initial vertical velocity in ft / sec (at t=0) Initial height in ft (at t=0) E 11: A soccer ball is kicked from ground level with an initial vertical velocity of 3 ft / s. After how many seconds will the ball hit the ground? h( t 6t t ) 1 vo ho h( t 16t t ) 3 0 The ball will hit the ground when its height is zero. 16t 3t 0 16 tt ( ) 0 16t 0 or ( t ) 0 t 0 or t The ball will hit the ground in seconds. Notice that the height is also zero when t 0, the instant that the ball is kicked. Check with calculator. Graph h( t) 16t 3t Notice zeros at 0 and. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 13 of 44 6/4/013

14 Using a Graphing Calculator to Solve Quadratic Equations E 1: Approimate the solution(s) of 1 4 using a graphing calculator. Step One: Write the equation in the form a b c 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 You Try: Solve the quadratic equation 5t 5 4t 6 by factoring. QOD: Solve the quadratic equation 6 3 graphically. Then check your answer algebraically. Closure: What must be true about a quadratic equation before you can solve it using the zero product property? Sample CCSD Common Eam Practice Question(s): What are the solutions of the quadratic equation A., B., C., 3 3 D. 4, ? Sample SAT Question(s): Taken from College Board online practice problems. If 6, which of the following must be true? (A) 6 (B) 3 (C) 0 (D) (E) Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 14 of 44 6/4/013

15 Sample Questions: 1. What is the correct factorization of? b. d.. What is the correct factorization of? b. d. ANS: C ANS: B 3. Factor by grouping. b. d. cannot be factored ANS: C 4. Factor. b. d. cannot be factored ANS: A 5. Factor. a. b. ANS: A 6. Determine whether is a difference of two squares. a. yes b. no ANS: B 7. Determine whether is a difference of two squares. If so, choose the correct factorization. a. yes; c. yes; b. yes; d. no ANS: B 8. Determine whether is a perfect square trinomial. If so, choose the correct factorization. a. yes; c. yes; b. yes; d. no ANS: D Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 15 of 44 6/4/013

16 9. Is completely factored? If not, what other factoring can occur? a. yes; the polynomial is completely factored. b. no; 4 can be factored from each term of the trinomial. c. no; the trinomial can be factored into two binomials. d. no; 4 can be factored from each term of the trinomial AND the resulting trinomial can be factored into two binomials. ANS: B 10. Completely factor. b. d. cannot be factored ANS: C 11. What is a quadratic function in standard form having zeros of 9 and? b. d. ANS: B 1. The height of an arrow that is shot upward at an initial velocity of 40 meters per second can be modeled by, where h is the height in meters and t is the time in seconds. Find the time it takes for the arrow to reach the ground. a. 8 sec c. sec b. 4 sec d. 6 sec ANS: A 13. What are the zeros of? a. 1, 5 c., b. d., 1, ANS: B 14. What are the zeros of? a., c., b., d., ANS: D 15. What are the zeros of the function? a., 1 c., b., 1 d., ANS: A Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 16 of 44 6/4/013

17 16. Find the zeros of the function. a. and 4 c. and 1 b. and 6 d. and 4 ANS: B 17. Find the zeros of. a. and c. 18 and b. and d. 18 and ANS: B 18. Find the zeros of. a. and 0 c. 0 and b. and d. and 10 ANS: D 19. Find the zeros of. a. or c. or b. or d. or ANS: D 0. What is a quadratic function in standard form having zeros of and? b. d. 1. What is a quadratic function in standard form having zeros of and 3? b. d.. What is a quadratic function in standard form having zeros of 5 and? b. d. ANS: B ANS: B ANS: C 3. Write a quadratic function in standard form having zeros of 3 and. b. d. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 17 of 44 6/4/013 ANS: A

18 4. Find the zeros of the function by factoring. a. or c. or b. or d. or 5. Find the roots of the equation by factoring. b. d. ANS: B ANS: C 6. Write a polynomial to represent the area of the shaded region. Then solve for given that the area of the shaded region is 4 square units b. d. ANS: B Short Answers: 1. Find the zeros of. ANS: 5 and. Find the zeros of. ANS: and Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 18 of 44 6/4/013

19 Essay 1. The equation gives the height h, in feet, of a football as a function of time t, in seconds, after it is kicked. How long does it take for the football to hit the ground? ANS: 3 s. The equation gives the height h, in feet, of a baseball as a function of time t, in seconds, after it is hit. How long does it take for the baseball to hit the ground? ANS: about.6 s 3. During a halftime show, a baton twirler releases her baton from a point 4 feet above the ground with an initial vertical velocity of 5 feet per second. Part A: Use the vertical motion model to write a function for the height seconds. (in feet) of the baton after Part B: Graph the function in Part A. Label the verte of the graph. Part C: How high does the baton go? Round your answer to the nearest tenth. Part D: How long after the baton is released does it reach its maimum height? Part E: At what moments is the baton at a height of 10 feet? Round your answer to the nearest hundredth. Part F: How much time does the twirler have if she plans to catch the baton on its way down at a height of 5 feet? Round your answer to the nearest hundredth. Part A: ANS: Part B: Part C: 13.8 ft Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 19 of 44 6/4/013

20 Part D: sec Part E: 0.30 sec and 1.7 sec after release Part F: 1.5 sec Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 0 of 44 6/4/013

21 Big Idea: One method of solving quadratic equations is by isolating the squared epression and finding the square root of both sides. Objectives: (Common Core) N.RN.A. Etend the properties of eponents to rational eponents. Rewrite epressions involving radicals and rational eponents using the properties of eponents. 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. Skill: Solve quadratic equations by finding square roots. Square Root: Radical Sign: Radicand: the number beneath the radical sign Properties of Square Roots a0, b 0 Product Property: a b a b Quotient Property: a b a b 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 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 Step Two: Find the square root of each pair of factors Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 1 of 44 6/4/013

22 E : Simplify the epression We must rationalize the denominator by multiplying by 1. 8 Now simplify the radical and the fraction Solving a Quadratic Equation by Finding Square Roots E 3: Solve the equation Step One: Isolate the squared epression Step Two: Find the square root of both sides Step Three: Solve for the variable. 6 or 6 E 4: Solve the equation n Step One: Isolate the squared epression. n 5 81 Step Two: Find the square root of both sides. Step Three: Solve for the variable. n 5 81 n 5 9 n 5 9 n 5 9 n 14 n 4 n 7 n Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page of 44 6/4/013

23 E 5: Solve the equation 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 6: 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 t 16 9 t t, 4 4 Solution: Since time is positive, the only feasible answer is seconds You Try: Solve the equation QOD: When is it necessary to simplify a square root? Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 3 of 44 6/4/013

24 Sample CCSD Common Eam Practice Question(s): Which of the following shows the solutions of 3 5? A. 8, a. 5,5 b. 4,4 c.,8 Sample CCSD Common Eam Practice Question(s): Use the formula below, where h is the height (in feet) of a falling object after t seconds and h is the object s initial height (in feet). 0 h 16t h0 A coyote is standing on a cliff 64 feet above a roadrunner. The coyote drops a boulder from the cliff. How much time does the roadrunner have to move out of its way? a. 1 4 second b. 1 second c. seconds d. 4 seconds Sample SAT Question(s): Taken from College Board online practice problems. If y 3, then (A) 4y (B) y (C) 4y (D) y (E) 4 y 6 must equal which of the following? Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 4 of 44 6/4/013

25 Sample Questions: 1. Solve. a. b.. Solve. a. b. ANS: A 3. Solve. ANS: A b. d. 4. Solve the equation. = 0 a. b. c. d. ANS: B DOK ANS: B 5. Solve. b. d. ANS: B 6. Solve the equation. a. 3 6 c. 3 6 b. 3 6 d. ANS: A 7. Marianna is making a piñata that has a ball-like shape. The piñata has a surface area of 60 square feet. Use the formula for the surface area of a sphere ( ) to find the radius of the piñata. a. About 4.78 ft c. About.80 ft b. About 3.87 ft d. About.19 ft ANS: D 8. Henry throws a tennis ball over his house. The ball is 6 feet above the ground when he lets it go. The quadratic function that models the height, in feet, of the ball after t seconds is. How long does it take for the ball to hit the ground? a. seconds c. 4.5 seconds Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 5 of 44 6/4/013

26 b. 3 seconds d. 6 seconds ANS: B 9. A pigeon lands on top of the Eiffel Tower and then, spotting a scrap of food, dives to the ground below. The pigeon s height in meters is approimately where t is the time in seconds. About how long is the pigeon in the air? a. 7.7 s c s b. 9.5 s d s ANS: A 10. The formula for finding the approimate volume of a cylinder is, where r is the radius and h is the height. The height of a cylinder is cm and the approimate volume is 9500 cm 3. Find the radius of the cylinder to the nearest hundredth of a centimeter. ANS: A ball bounces straight up from the ground with an initial vertical velocity of 8 feet per second. Use a vertical motion model to write and solve an equation to find the number of seconds it takes for the ball to return to the ground. Eplain your reasoning. ANS: 0.5 seconds; The height h, in feet, of the ball after t seconds is given by the equation. When the ball returns to the ground, the height will be 0, so Factor the right side of the equation to get The solutions of the equation are, or t = 0, and, or t = 0.5. The solution t = 0 represents the time at which the ball first bounces, so the other solution, t = 0.5, represents the number of seconds it takes for the ball to return to the ground. 1. A soccer ball is resting on the ground. Kendra kicks the ball. Its initial vertical velocity is 30 feet per second. Part A: Write an equation that models the height h of the ball t seconds after Kendra kicks it. Part B: Suppose the ball hits the ground without being caught. Use a graphing calculator to determine the approimate number of seconds the ball was in the air. Justify your answer. Part C: Suppose the ball reaches its highest point, then starts to fall back to the ground. Before it hits the ground, it bounces off the head of another player at a point that is 5 feet from the ground. Use a graphing calculator to determine the approimate number of seconds the ball was in the air before it bounced off the player's head. Justify your answer. ANS: Part A: Part B: about 1.9 seconds; Use a graphing calculator to graph. Then find the value of when y = 0 by finding the - coordinates of the points where the graph intersects the -ais. You can see that y = 0 when = 0 seconds and when = seconds. The solution = 0 represents the time t when Kendra kicked the ball. The other solution represents the number of seconds after which the ball returned to the ground. So, the ball was in the air about 1.9 seconds before it returned to the ground. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 6 of 44 6/4/013

27 Part C: about 1.7 seconds; Use the graph of to find the value of when y = 5. There are two points with this y-coordinate on the graph. The -coordinates of these points are approimately 0.18 and 1.7. The first point, where, represents the first time after Kendra kicks the ball that it is at a height of 5 feet. The second point, where, represents the time when the ball is returning to the ground and it bounces off the player's head. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 7 of 44 6/4/013

28 Big Ideas: The Square Root Property can be used to solve an equation of the form a b c d when a b c is a perfect square trinomial. If the trinomial is not a perfect square, a method called completing the square can be used to rewrite the equation so that the trinomial is a perfect square. Objectives: (Common Core) 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. Skill: solve quadratic equations by factoring, graphing, completing the square, and the quadratic formula. Review: Factoring a Perfect Square Trinomial a ab b a b a ab b a b 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. b as a perfect square trinomial in order Teacher Note: Algebra Tiles work well to illustrate completing the square. See Page 81 for an activity. E 1: Find the value of c such that 10 c is a perfect square trinomial. b 10, therefore we must add Note: 10 5 c to complete the square. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 8 of 44 6/4/013

29 Solving a Quadratic Equation by Completing the Square E : Solve 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) Step Four: Factor the perfect square trinomial Step Five: Take the square roots of both sides. Step Si: Solve for the variable E 3: Solve The solution set is 3 11, 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). Step Four: Factor the perfect square trinomial Step Five: Take the square roots of both sides. Step Si: Solve for the variable i 7 i 7 i 7 i 7 i The solution set is 7 i,7 i Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 9 of 44 6/4/013

30 Verte Form of a Quadratic Function: y a h k Verte: hk, E 4: Write the quadratic function in verte form and identify the verte of y 4 7. Step One: Factor out the lead coefficient from the variable terms (if other than one). y 7 Step Two: Complete the square. b Note: We must add a to both sides by the distributive property. y 7 y 1 7 y 1 7 Step Three: Factor the perfect square trinomial. Step Four: Solve for y. y 1 7 y 1 9 The verte is 1, 9. E 5: Write the quadratic function in verte form and identify the verte of y 5. Step One: Factor out the lead coefficient (if other than one). y 5 Step Two: Complete the square. 5 5 y y Step Three: Factor the perfect square trinomial. 5 5 y 4 Step Four: Solve for y. 5 5 y y 4 The verte is 5 17, 4. You Try: Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 30 of 44 6/4/013

31 1. A rectangle has sides and 10. The area of the rectangle is 100. Use completing the square to find the value of.. Write the quadratic function the function? y in verte form. What is the maimum value of QOD: Why is completing the square helpful when finding the maimum or minimum value of a quadratic function? Sample CCSD Common Eam Practice Question(s): 4. What are the solutions of 610 0? d. = or = 4 e. = 10 or = 4 f. = 3 + i or = 3 i g. = 3 + i or = 3 i 1. Which is one of the appropriate steps in finding solutions for completing the square? when A. 4 3 h. 3 i. 4 7 j. 7 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 31 of 44 6/4/013

32 Sample Questions: 1. Determine whether is a perfect square trinomial. If so, choose the correct factorization. a. yes; c. yes; b. yes; d. no ANS: D. Solve by completing the square. a. 3 and c. 3 and 0 b. 6 and d. 6 and 3 ANS: B 3. A gardener wants to create a rectangular vegetable garden in a backyard. She wants it to have a total area of 10 square feet, and it should be 1 feet longer than it is wide. What dimensions should she use for the vegetable garden? Round to the nearest hundredth of a foot. a feet by feet c feet by 4.49 feet b feet by feet d feet by.95 feet ANS: A 4. Solve the equation by completing the square. a. = or = c. = 1 or = 3 b. = 1 or = 3 d. = or = 6 ANS: B 5. Complete the square for the epression. Write the resulting epression as a binomial squared. b. d. ANS: A 6. Solve by completing the square: = 0 a. -4, 6 c. -4, -6 b. 4, 6 d. 4, -6 ANS: A 7. Solve by completing the square: a and 3 11 c and 3 11 b and 3 7 d and 3 7 ANS: C 8. Solve by completing the square. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 3 of 44 6/4/013

33 b. d. ANS: D 9. Complete the square for. Then write the resulting epression as a binomial squared. a. 49; 7 c. 49; 7 b. 49; 7 d. 49; 7 ANS: C 10. Find the positive root of the equation by completing the square. ANS: Solve the equation by completing the square. = 0 a. 8, b. 8, c. 8, d. 8, ANS: B 1.The hypotenuse of a right triangle is 13 cm. One of the legs is 7 cm longer than the other leg. Find the area of the triangle ANS: 30 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 33 of 44 6/4/013

34 Big Idea: The Quadratic Formula can be used to solve any equation in the form a b c 0, where a 0. To determine the roots of the equation, substitute the coefficients a and b and the constant c into the quadratic formula and then simplify the resulting epression. Objectives: (Common Core) A.CED.A.1 Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and eponential functions. A.SSE.A. 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. Skills: Solve quadratic equations by quadratic formula Analyze the nature of the roots of a quadratic equation. Deriving the Quadratic Formula by Completing the Square E 1: 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. a b c 0 a a a a b c 0 a a b c a a Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 34 of 44 6/4/013

35 b Step Three: Complete the square (add to both sides). b b c b a a a a b b 4ac b a 4a 4a b b 4ac Step Four: Factor the perfect square trinomial. a 4a b b 4ac a 4a Step Five: Take the square roots of both sides. b b 4ac a 4a b b 4ac b b 4ac a a a a Step Si: Solve for the variable. b b 4ac b b 4ac a a 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 : Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessary) 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 Step Four: Simplify The solution set is 4 15,4 15 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 35 of 44 6/4/013

36 E 3: Solve the quadratic equation 3 7 using the quadratic formula. Step One: Rewrite in standard form (if necessary) Step Two: Identify a, b, and c. a, b 3, c 7 Step Three: Substitute the values into the quadratic formula. Step Four: Simplify. The solution set is i, i b b 4ac a i i Discriminant: The number under the square root in the quadratic formula. b 4ac The sign of the discriminant determines the number and type of solutions of a quadratic equation. If If If b b b 4ac 0, then the equation has two real solutions (two -intercepts). 4ac 0, then the equation has one real solution (one -intercept). 4ac 0, then the equation has two imaginary solutions (no -intercept). E: What is the discriminant of the quadratic equation ? Give the number and type of solutions the quadratic equation has. Then graph the quadratic function y to verify your answer. Discriminant: b ac Since the disciminant is 0, there is one real solution. The -coordinate of the verte of the function y is 1 1 The y-coordinate of the verte is y Plot a couple of other points to graph the parabola. Note that the graph has one -intercept (the verte). Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 36 of 44 6/4/013 b 4 1. a 4

37 You Try: Determine the number and type of solutions the quadratic equation has. Then solve the equation using the quadratic formula. 0n 6n 6n 13n 3 QOD: Solve the equation a b c 0 by completing the square. Sample CCSD Common Eam Practice Question(s): 1. How many real and imaginary solutions are there for the equation ? B. no real solutions, imaginary solutions k. 1 real solution, no imaginary solutions l. 1 real solution, 1 imaginary solution m. real solutions, no imaginary solutions. What is the solution set for the quadratic equation A. 3 3, 3 3 n. 3 6, 3 6 o. 3 3,3 3 p. 3 6, ? 3. What are the solutions of ? q. r. s. t. i i Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 37 of 44 6/4/013

38 Sample Questions: 1. Use the Quadratic Formula to solve. b. d. ANS: C DOK 1. Use the Quadratic Formula to solve. b. d. ANS: B DOK 1 3. Use the Quadratic Formula to solve. b. d. ANS: A DOK 1 4. Solve using the Quadratic Formula. a. = 18 or = c. = 9 or = 1 b. = 54 or = 46 d. = 1 or = 9 ANS: C DOK 5. A farmer has 10 yards of fencing to build around a rectangular orchard. Let w be the width of the orchard. Write an equation giving the area of the orchard. Find the dimensions of the orchard when the area is 900 square yards. a. ; The width is 58 yd, and the length is 6 yd. b. ; The width is 50 yd, and the length is 18 yd. c. ; The width is 30 yd, and the length is 30 yd. d. ; The width is 7 yd, and the length is 48 yd. ANS: C DOK 6. During the eruption of Mount St. Helens in 1980, debris was ejected at a speed of over 440 feet per second (300 miles per hour). The height in feet of a rock ejected at angle of 75 is given by the equation, where t is the time in seconds after the eruption. The rock s horizontal distance in feet from the point of ejection is given by. Assuming the elevation of the surrounding countryside is 0 feet, what is the horizontal distance from the point of ejection to the where the rock would have landed? Round your answer to the nearest foot. a.,34 ft c. 4,467 ft. b. 8,93 ft d. 1,117 ft ANS: C DOK Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 38 of 44 6/4/013

39 7. The number of new cars purchased in a city can be modeled by the equation where C is the number of new cars and t is the number of years since In what year will the number of new cars reach 15,000? a. 06 b c d ANS: D DOK 8. Solve. a., c., b., d., ANS: C DOK 9. Find the zeros of the function. a. = 3 or 3 c. = b. = or 3 3 d. = or 6 3 ANS: B DOK 10. Find the zeros of the function. b. d. ANS: D DOK Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 39 of 44 6/4/013

40 Big Idea: Graphing quadratic inequalities is similar to graphing linear inequalities. After graphing test a point inside the parabola. If the point is a solution shade inside, otherwise, the region outside is shaded. Solving a quadratic inequality algebraically is similar to solving a linear inequality. The difference is many quadratic inequalities have not one but two solutions. The number line is divided into three intervals. Testing a value from each interval on the number line reveals which solution set(s) are correct. Objectives: (Common Core) A.CED.A.1 Create equations that describe numbers or relationships. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and eponential functions. A.CED.A.3 Create equations that describe numbers or relationships. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling contet. A.REI.B.3 Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Skill: The student will graph and solve quadratic inequalities with and without technology. Graphing a Quadratic Inequality in Two Variables E 1: Graph the quadratic inequality y 5 6. Step One: Graph the parabola. Make the parabola dashed if < or > and solid if or. We will write the inequality in verte form using completing the square. y 6 y1 1 6 y Note: We draw a solid parabola. y 1 7 Step Two: Choose a test point inside the parabola and substitute it into the inequality. We will choose y 6 0, true Step Three: If the test point makes the inequality true, shade inside the parabola. If it does not, shade outside the parabola. Graphing Calculator Activity Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 40 of 44 6/4/013

41 E : Graph the quadratic inequality y 9 by hand and then check your graph on the graphing calculator using the Inequalz Application. Step One: Graph the parabola. Make the parabola dashed if < or > and solid if or. The verte of the parabola is the point 0,9. Note: We draw a dashed parabola that opens down. Step Two: Choose a test point inside (not on) the parabola and substitute it into the inequality. We will choose 0,0. y false Step Three: If the test point makes the inequality true, shade inside the parabola. If it does not, shade outside the parabola. We will shade outside the parabola. To check your graph, turn on the application by choosing Inequalz after pressing the APPS key. Press any key, and now your Y= screen should look like this: Enter the function y 9 into Y1. Then use the command (function) buttons along the bottom of your calculator screen to choose >. Note In order to use the command buttons, you must first type the ALPHA key. So to choose >, we will press ALPHA TRACE. Graph the inequality. (For the graph shown, we used ZOOM STANDARD). Solving a Quadratic Inequality in One Variable E 3: Solve Step One: Solve the quadratic equation 6 0 using any method. 3 0 We will use factoring Step Two: Draw a sign chart on a number line to test which values for satisfy the inequality. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 41 of 44 6/4/013

42 Choose an -value to the left of and substitute into the inequality. We will try true Choose an -value between and 3 and substitute into the inequality. We will try false Choose an -value to the right of 3 and substitute into the inequality. We will try true Step Three: Write the solution as a compound inequality. or 3 You Try: 1. Graph the quadratic inequality y 3.. Solve the quadratic inequality QOD: What is the purpose of a sign chart when solving a quadratic inequality in one variable? Sample CCSD Common Eam Practice Question(s): Which of the following graphs represents the quadratic inequality y 4? Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 4 of 44 6/4/013

43 Sample: 1. Solve the inequality by using algebra. b. d. ANS: C DOK 1. Solve the inequality. b. d. ANS: D DOK 1 3. The daily profit P for a cake bakery can be modeled by the function, where is the price of a cake. What should the price of a cake be to provide a daily profit of at least $600? Round your answer(s) to the nearest dollar. b. d. ANS: B DOK Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 43 of 44 6/4/013

44 The Big Idea: Quadratic models can be used to represent data. Objective: (Common Core) A.CED.A. Create equations that describe numbers or relationships. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales A.CED.A.3 Create equations that describe numbers or relationships. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling contet. For eample, represent inequalities describing nutritional and cost constraints on combinations of different foods Skills: The student will develop mathematical models involving quadratic equations to solve real-world problems. Identifying Quadratic Data Determine whether each data set could represent a quadratic function. Eplain y Find the first and second differences. Check that the values are equally spaced y st : nd : Quadratic function; second differences are constant for equally spaced values. Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 44 of 44 6/4/013

45 Graphing Calculator Activity: Using a quadratic model to represent data. E: The table shows the average sale price p of a house for various years t since Use a quadratic regression on the graphing calculator to write a quadratic model for the data. Years Since 1988, t Average Sale Price (thousands of dollars), p Enter the data from the table into the Lists. Enter t values into L1 and p values into L. (Use STAT Edit to enter data into the Lists.) On the home screen, use the QuadReg to find the quadratic regression. (Use STAT Calc to find QuadReg.) Note: To store this into Y1, you can type in Y1 after QuadReg on the home screen before pressing enter. Keystrokes for entering Y1: Take a look at the graph of the quadratic model with the scatter plot of the data. Sample SAT Question(s): Taken from College Board online practice problems. If y and y 8, what is the value of? (A) 1 (B) (C) 4 (D) 8 (E) 16 Alg II Unit 05d Quadratic Functions - Intercepts & Solutions CCSS (Repaired) Page 45 of 45 6/4/013

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