x Radical Sign: Radicand: the number beneath the radical sign

Transcription

1 Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. Square Root: Radical Sign: Radicand: the number beneath the radical sign Positive (Principal) Square Root: 9 3 Negative Square Root: 9 3 Review: Simplifing 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: Simplif 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. 3 3 E: Simplif the epression We must rationalize the denominator b multipling b 1. 8 Now simplif the radical and the fraction Quadratic Equation: an equation that can be written in the standard form a b c 0, a 0 Page 1 of 33 McDougal Littell: 9.1, , , 1.4

2 Solving a Quadratic Equation b Finding Square Roots: to use this method, the quadratic equation must be of the form a c 0, b 0 E: Solve the equation 16. Step One: Find the square root of both sides Step Two: Solve for. (Note that there are two solutions.) 4, 4 This can also be written 4 E: Solve the equation 4 4. Step One: Isolate the squared epression Step Two: Find the square root of both sides. 0 0 Step Three: Solve for. (Note that there is one solution.) 0 E: Solve the equation 3. Step One: Isolate the squared epression. 3 1 Step Two: Find the square root of both sides is not a real number Step Three: Solve for. (Note that there is no solution.) This equation has no real solution. E: Solve the equation Step One: Isolate the squared epression Page of 33 McDougal Littell: 9.1, , , 1.4

3 Step Two: Find the square root of both sides Step Three: Solve for the variable. 6 or 6 E: Solve the equation Step One: Isolate the squared epression. Step Two: Find the square root of both sides Step Three: Solve for the variable. 5 or 5 E: 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 59 n5 9 n 14 n 4 n 7 n E: Solve the equation a, Step One: Isolate the squared epression. a 8 8 Step Two: Find the square root of both sides. a 8 8 a 8 7 Page 3 of 33 McDougal Littell: 9.1, , , 1.4

4 Step Three: Solve for the variable. a8 7 a8 7 a 8 7 a 8 7 a 8 7 Note: The ( plus or minus ) smbol is used to write both solutions in a shorter wa. 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 b the function h 16t h0, where h0 is the initial height of the object. E: 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. h 16t h0 h0 81, h 0 Initial height is 81 ft. The ball will hit the ground when its height is 0 ft. Solve for t. 016t 81 16t t 16 9 t t, 4 4 Solution: Since time is positive, the onl feasible answer is seconds You Tr: Solve the equation QOD: Wh do some quadratic equations have two, one, or no real solution? Page 4 of 33 McDougal Littell: 9.1, , , 1.4

5 Sample CCSD Common Eam Practice Question(s): 1. Solve the equation A. B. C. D , 5 5, 4 4 Sample Nevada High School Proficienc Eam Questions (taken from 009 released version H): An equation is shown below What is the solution set of the equation? A 5 B 5,5 C 5 D 1.5,1.5 Page 5 of 33 McDougal Littell: 9.1, , , 1.4

6 Sllabus Objectives: 9. The student will compare characteristics of a given famil of quadratic functions. 9.3 The student will determine the domain and range of quadratic equations algebraicall and graphicall. Quadratic Function: a function that can be written in the form Parabola: the U-shaped graph of a quadratic function a b c, when a 0 Verte: the highest or lowest point on a quadratic function (maimum or minimum) Ais of Smmetr: the vertical line that passes through the verte of a quadratic function Verte Ais of Smmetr Verte Ais of Smmetr 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 greater than or equal to the - coordinate of the verte. If the parabola opens down, then the range is all values of less than or equal to the - coordinate of the verte. Page 6 of 33 McDougal Littell: 9.1, , , 1.4

7 Graphing a Parabola E: Graph the quadratic function. Step One: Make a table of values (t-chart) 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 smmetr is 0. E: Graph the parabola Step One: Make a table of values (t-chart) 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 smmetr is Comparing and : The verte is 0,0, and the ais of smmetr is 0 for both graphs. When a is positive, the parabola opens up; when a is negative, the parabola opens down. Page 7 of 33 McDougal Littell: 9.1, , , 1.4

8 Activit: Transformations with and. Use the graphing calculator to graph the quadratic functions. Describe the effect on the graphs of and dotted line.). (Note: In the calculator graphs shown, or is graphed as a 1. Compare to Verte: Same 0,0 Opens narrower than. 1 3 Compare to Verte: Same 0,0 Opens wider than 3. 3 Compare to Verte: Up 3 0,3 Opens the same as 4. 4 Compare to Verte: Down 4 0, 4 Opens the same as Conclusions (sample): For quadratic functions of the form a c a 1 Opens down Narrower than 1a 0 Opens down Wider than 0a 1 Opens up Wider than a 1 Opens up Narrower than c 0 Verte moves down c units c 0 Verte moves up c units Page 8 of 33 McDougal Littell: 9.1, , , 1.4

9 Standard Form of a Quadratic Function: a b c Verte: the -coordinate of the verte is b Ais of Smmetr: a b a Graphing a Quadratic Function in Standard Form E: Graph the quadratic function and range State the verte, ais of smmetr, domain, Step One: Find the -coordinate of the verte. a 1, b 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 coordinate of verte: Note: When calculating the -coordinate of points to the right and left of the verte, notice the smmetr. Step Three: Plot the points from the table and draw the parabola. 10 Verte: 3, 10 Ais of Smmetr: Domain: all real numbers -10 Range: 10 Page 9 of 33 McDougal Littell: 9.1, , , 1.4

10 E: Graph the quadratic function 1 1. State the verte and ais of smmetr. 3 Step One: Find the -coordinate of the verte. 1 a, b 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, ou ma want to choose values that will guarantee whole numbers for the -coordinates.) Step Three: Plot the points from the table and draw the parabola Verte: 3, 4 Ais of Smmetr: 3-10 E: Graph the quadratic function Step One: Find the -coordinate of the verte. a, b 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 coordinate of verte: Note: When calculating the -coordinate of points to the right and left of the verte, notice the smmetr. Step Three: Plot the points from the table and draw the parabola Verte:, 18 Ais of Smmetr: Page 10 of 33 McDougal Littell: 9.1, , , 1.4

11 Using a Quadratic Model E: A basketball s path can be modeled b , where represents time (in seconds) and 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 -coordinate of the verte, which is approimatel 9.5ft. You Tr: Find the verte and ais of smmetr for the following quadratic function. Determine if the parabola will open up or down. Then graph the parabola. QOD: How man points does it take to determine a unique parabola? Page 11 of 33 McDougal Littell: 9.1, , , 1.4

12 Sample CCSD Common Eam Practice Question(s): 1. Which of the following is the graph of 6? A. B. C. D. Page 1 of 33 McDougal Littell: 9.1, , , 1.4

13 . Which equation best represents the following graph? A. B. C. D Which of the following are true statements about the graph of 8 4? I. Opens Up II. Opens Down III. Ais of smmetr = 4 IV. Ais of smmetr = 4 B. I and III onl C. I and IV onl D. II and III onl E. II and IV onl Page 13 of 33 McDougal Littell: 9.1, , , 1.4

14 4. Find the verte of the parabola: A. 6, 7 B. 3, 11 C. 3, 61 D. 6, What is the domain and range of the function below? 4 shown in the graph A. Domain: all real numbers Range: all real numbers B. Domain: Range: 4.5 C. Domain: all real numbers Range: 4.5 D. Domain: Range: all real numbers Page 14 of 33 McDougal Littell: 9.1, , , 1.4

15 Sample Nevada High School Proficienc 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 ()? Page 15 of 33 McDougal Littell: 9.1, , , 1.4

16 Sllabus Objectives: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. 9.5 The student will graph quadratic equations and find possible solutions to those equations using coordinate geometr. Solving a Quadratic Equation b Graphing Step One: Write the equation in the form a b c 0. Step Two: Graph the function 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 interchangeabl for the above value. E: Solve the equation 1 8 b graphing. Then check b solving algebraicall. Step One: Write the equation in the form a b c Step Two: Graph the function a b c. 1 8 Verte: b 0 0 a Step Three: Find the -intercept(s) of the function. 0, 8 The zeros are at 4 and 4, so the solutions are 4, Solve algebraicall: Page 16 of 33 McDougal Littell: 9.1, , , 1.4

17 E: Solve the equation 3 b graphing. Step One: Write the equation in the form 3 0 a b c Step Two: Graph the function a b c. 3 5 Verte: b 3 3 a , Step Three: Find the -intercept(s) of the function. The zeros are at 1 and, so the solutions are 1, E: Solve the equation 4 4 b graphing. -10 Step One: Write the equation in the form 0. a b c 44 0 Step Two: Graph the function 4 4 a b c. 10 Verte: b 4 4 a , Step Three: Find the -intercept(s) of the function. The root is at, so the solution is Check: Page 17 of 33 McDougal Littell: 9.1, , , 1.4

18 E: Solve the equation 3 0 graphicall. Step One: Write the equation in the form 0. a b c 3 0 Step Two: Graph the function 3 Verte: b a a b c. 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: 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 a b c. 4 1 Step Three: Find the zero(s) of the function. 4.36,0.36 You Tr: Solve the quadratic equation 6 3 graphicall. Then check our answer algebraicall. QOD: How can ou tell from the graph of a quadratic function if the equation has one, two, or no solution? Page 18 of 33 McDougal Littell: 9.1, , , 1.4

19 Sample CCSD Common Eam Practice Question(s): The graph of 1 has how man -intercepts? A. 1 B. C. 1 D. 0 Sample Nevada High School Proficienc 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 b the equation below 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 Page 19 of 33 McDougal Littell: 9.1, , , 1.4

20 Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. Review: Factoring Quadratic Trinomials into Two Binomials (Using the ac method or splitting the middle term.) Factoring, 1 a b c a E: 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, 1 a b c a E: Factor 7 3. Step One: Multipl a c. 3 6 Step Two: Find two integers such that their product is a c 6 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 Step Four: Factor b grouping. Group the first terms and last terms and factor out the GCF from each pair Step Five: If Step Four was done correctl, 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: Multipl a c Step Two: Find two integers such that their product is a c 10 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. Page 0 of 33 McDougal Littell: 9.1, , , 1.4

21 5 5 Step Four: Factor b grouping. Group the first terms and last terms and factor out the GCF from each pair Step Five: If Step Four was done correctl, 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 aba b Perfect Square Trinomial: a abb ab a abb ab Zero Product Propert: If the product of two factors is 0, then one or both of the factors must equal 0. E: Solve the equation using the zero product propert. Since one or both of the factors must equal 0, we will solve the two equations 30 and Solutions: 1, 3 Solving a Quadratic Equation b Factoring E: Solve the equation 5 6 b factoring. Step One: Write the equation in standard form Step Two: Factor the quadratic. 3 0 Step Three: Set each factor equal to zero and solve. Note: Check this answer b graphing on the calculator , 3 Page 1 of 33 McDougal Littell: 9.1, , , 1.4

22 E: 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 E: Solve the equation Step One: Write the equation in standard form Step Two: Factor the quadratic Note: Step Three: Set each factor equal to zero and solve. The solution can be written in set notation: 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 b plotting the zeros. The verte is halfwa between the zeros. E: Find the zero(s) of the quadratic function Step One: Factor the quadratic polnomial. 3and graph the parabola Page of 33 McDougal Littell: 9.1, , , 1.4

23 Step Two: Set each factor equal to 0 and solve. Step Three: Find the coordinates of the verte. Step Four: Plot the points and sketch the parabola You Tr: Solve the quadratic equation 5t 5 4t 6 b factoring. QOD: What must be true about a quadratic equation before ou can solve it using the zero product propert? Sample CCSD Common Eam Practice Question(s): 1. What is the solution set for the following equation? A. { 9, 1} B. { 9, 1} C. { 1, 9} D. {1, 9}. Which of the following equations has roots of 7 and 4? A B C D Page 3 of 33 McDougal Littell: 9.1, , , 1.4

24 Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. Review: Factoring a Perfect Square Trinomial a abb ab a abb ab Completing the Square: writing an epression of the form to factor it as a binomial squared b as a perfect square trinomial in order To complete the square of b, we must add b. Teacher Note: Algebra Tiles work well to illustrate completing the square. See Page 79 for an activit. E: 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. Solving a Quadratic Equation b Completing the Square E: Solve 87 0 b 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 The solution set is 7, 1. Check our answer b factoring. E: Solve 14 0 b completing the square. Page 4 of 33 McDougal Littell: 9.1, , , 1.4

25 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. The solution set is 3 11, E: Solve 3 0 b 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. Step Five: Take the square roots of both sides. Step Si: Solve for the variable The solutions are. to both sides) Page 5 of 33 McDougal Littell: 9.1, , , 1.4

26 You Tr: Solve b completing the square QOD: Describe wh 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, ? C. D , , Page 6 of 33 McDougal Littell: 9.1, , , 1.4

27 Sllabus Objective: 9.4 The student will solve quadratic equations using graphic and algebraic techniques to include the quadratic formula, square roots, factoring, completing the square, and graphing. Deriving the Quadratic Formula b Completing the Square Solve the quadratic equation a b c 0 b completing the square. Step One: Rewrite so that the lead coefficient is 1. a b c 0 a a a a b c 0 a a Step Two: Take the constant term to the other side. b c a a b Step Three: Complete the square (add to both sides). b b c b a a a a b b 4acb 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. Page 7 of 33 McDougal Littell: 9.1, , , 1.4

28 E: Solve the quadratic equation 8 1 using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a 1, b8, c 1 Step Three: Substitute the values into the quadratic formula. b b 4ac a Step Four: Simplif The solution set is 4 15,4 15 E: Solve the quadratic equation using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a 6, b5, c 1 Step Three: Substitute the values into the quadratic formula. Step Four: Simplif. The solution set is 1 1, 6 b b 4ac a E: Solve the quadratic equation using the quadratic formula. Step One: Rewrite in standard form (if necessar) Step Two: Identif a, b, and c. a 9, b1, c 4 Page 8 of 33 McDougal Littell: 9.1, , , 1.4

29 b b 4ac a Step Three: Substitute the values into the quadratic formula Step Four: Simplif The solution set is 3. E: Solve the quadratic equation 3 0 using the quadratic formula. Step One: Rewrite in standard form (if necessar). 3 0 Step Two: Identif a, b, and c. a, b, c 3 Step Three: Substitute the values into the quadratic formula. b b 4ac a Step Four: Simplif. 4 4 There is no real solution to the quadratic equation because 0 is not a real number. You Tr: 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. Page 9 of 33 McDougal Littell: 9.1, , , 1.4

30 Sample CCSD Common Eam Practice Question(s): 1. What are the roots (solutions) of A. 1 3, 1 3 B. 1 5, ? C. D , ,. 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 , , , , Page 30 of 33 McDougal Littell: 9.1, , , 1.4

31 Sllabus Objective: 9.6 The student will solve practical problems involving quadratic equations with a variet of methods, including discrete methods (with and without technolog). Discriminant: the discriminant of the quadratic equation 0 is a b c 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: Determine the number of real solutions that the equations have Rewrite the equation in standard form a 3, b1, c 1 Find the discriminant. b ac Determine the number of real solution(s). b 4ac 11 0, so there are no real solutions.. 45 Rewrite the equation in standard form a 1, b5, c 4 Find the discriminant. b ac Determine the number of real solution(s). b 4ac 41 0, so there are two real solutions Rewrite the equation in standard form. a 9, b1, c Find the discriminant. b ac Determine the number of real solution(s). b 4ac 0, so there is one real solution. Page 31 of 33 McDougal Littell: 9.1, , , 1.4

32 Determining the Number of -Intercepts of a Quadratic Function Using the Discriminant Because the -intercepts of 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: Sketch the graph of a quadratic function with a negative discriminant. Because the discriminant, b 4ac 0, the function will have no -intercept. 10 A sample answer is shown in the graph. Note: An parabola which does not intersect the -ais is an acceptable answer. Application Problem E: 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 velocit of 45 feet per second. This can be modeled b 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 t 45t t 45t 34 a 16, b45, c 34 Find the discriminant. b ac 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 Find the maimum (verte). The baton will reach approimatel ft. You Tr: Find values for c so that the equation will have no real solution, one real solution, and two real solutions. 3c 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 polnomial is factorable using the discriminant. (It must be a perfect square.) Page 3 of 33 McDougal Littell: 9.1, , , 1.4

33 Sample CCSD Common Eam Practice Question(s): The graph of 1 has how man -intercepts? A. 1 B. C. 1 D. 0 Page 33 of 33 McDougal Littell: 9.1, , , 1.4