x 2 + x + x 2 x 3 b. x 7 Factor the GCF from each expression Not all may be possible. 1. Find two numbers that sum to 8 and have a product of 12

Transcription

1

2 Factor the GCF from each expression x 3x. 16x 4 Name: a. b x y 36x y 4x y x x 3 x 3 c. d. Not all may be possible. 1. Find two numbers that sum to 8 and have a product of 1. Find two numbers that sum to 5 and have a product of 6 3. Find two numbers that sum to 5 and have a product of Find two numbers that sum to -6 and have a product of 1 5. Find two numbers that sum to 16 and have a product of Find two numbers that sum to -4 and have a product of Find two numbers that sum to 1 and have a product of Find two numbers that sum to -14 and have a product of Find two numbers that sum to 0 and have a product of Find two numbers that sum to 8 and have a product of Multiply the following: x 3 b. x 7 x a. x 6 x + x + x + x + Notice: What is the sum of the constants in each binomial above? Notice: What is the product of the constants in each binomial above? Notice: What is the sum of the constants in each binomial above? Notice: What is the product of the constants in each binomial above? M. Winking Unit 3-1 page 74

3 1. FACTOR the following (not all may be factored): a. x 9x 18 b. 6x 40 x c. x 5x 14 d. a 7a 6 d. m 8m 16 e. g 11g 4 f. x 5x 6 g. x 5x 6 h. m 7m 60 i. g 14g j. 3x 4x 60x k. 5x 5x 30x M. Winking Unit 3-1 page 75

4 13. Special Forms Name Formula Example Difference of two A B A B A B 64x 9 8x 3 8 squares Perfect square trinomials x 3 8x 3 AB B A AB B A x 14x 49 x x 7 7 x 7 A B A B a. x 36 b. m 9 c. m 4 81 d. 4b 400 e. 4x 1x 9 f. 64a 48a 9 g. 11 b 8 4 a h. 18m 48m 3m i x 60x y 5y 14. Find the volume of the rectangular prism shown below 15. Describe the area of the shaded region as a polynomial M. Winking Unit 3-1 page 76

5 15. Multiply the following: 3 x 1 a. x b. 4x 3 x FACTOR the following: a. 6x 1x b. 4x 9x 9 c. x 7x 15 d. 3a 10a 8 e. 5g 14g 8 f. 6m 10m 4 3 g. 6b 8b 30b h. 5m 11m 1 M. Winking Unit 3-1 page 7

6 Unit 3 Name: Solve the following QUADRATIC EQUATIONS using the SQUARE ROOT METHOD: 1. w y m b x a Solve the following QUADRATIC EQUATIONS by FACTORING & ZERO PRODUCT PROPERTY: 1. w w 4. t 8t 0 3. r 5 6r x 5. x ( x 4) 4 x 4. x 0 7x 4 6. x 5 3x 11 10x M. Winking Unit 3- page 78

7 (Continued) Solve the following QUADRATIC EQUATIONS by FACTORING & ZERO PRODUCT PROPERTY: 7. x 7x p 10 3p 9. 3x x 1 3x 6x x x x x a 17a 8 0 Solve the applications that of QUADRATIC EQUATIONS: 1. The length of a rectangle is 1 cm more than twice its width. If the area of the rectangle is 1cm then what are the dimensions?. The length of a rectangle is 3 cm less than twice its width. If the area of the rectangle is 0cm then what are the dimensions? 3. The volume of the prism is 8 cubic inches. What is the length of each side? 4. The volume of the prism is 45 cubic inches. What is the length of each side? M. Winking Unit 3- page 79

8 5. A picture frame is shown at the right. If the entire area of the frame and the picture totals 10 square inches find the width of the frame. 6. A below ground swimming pool is to be constructed in the park. The pool is in the shape of a rectangle with the dimensions of 0 by 4. A uniform width sidewalk is to be made around the pool. If the contractor says that he has enough concrete to create 300 ft of sidewalk. What is the maximum width of the sidewalk around the pool? 7. The product of two consecutive positive integers is 13. Write an equation to model the situation and find the two integers. 8. The perimeter of a rectangle is 4 cm and the area is 80 cm. Write an equation to model the situation and find the dimensions of the rectangle. 9. A park is putting in a sidewalk of uniform width to go around two sides of a rectangular garden that is 10 feet by 30 feet. The contractor has enough concrete for 176 ft. What is the maximum width of such a sidewalk? M. Winking Unit 3- page 80

9 10. A right triangle is shown below. Use the Pythagorean Theorem to determine the lengths of each side.

10 BASIC x 10x x 6 0 INTERMEDIATE 3x 15x 9 0 x Move constant to the other side x 10x 6 x 10x ½ of x 10x 5 31 Easily Factors 5 x 5 31 x Can be re-written x 5 31 Take the square root of both sides x 5 31 Don t forget. x 5 31 Isolate x 3 15x 9 0 x 5 31 OR x or x Move constant to the other side x 9 x Divide both sides by the leading coefficients 3x 15x x 5 5 5x Square 5 3 Add 5 to both sides x 5x ½ of Square 5/ Notice it is the number that when added to itself is equal to 5 and when multiplied by 5 x x x 13 4 itself is equal to. 5 x 13 4 x 5 13 x 5 13 x 5 13 x or M. Winking Unit 3-3 page 81

11 Solve the following by completing the square x 19 0 x. x 1x 0 3. x 1x x 5x x 14 0 x 6. 3x 18x 5 0 M. Winking Unit 3-3 page 8

12 Solve the following by completing the square. 7. x 16x x 10x x 5x x 10x 8 0 M. Winking Unit 3-3 page 83

13 Solve the following by completing the square. Derive the quadratic formula. a x b x c 0 b x b 4ac a M. Winking Unit 3-4 page 84

14 The Quadratic Formula Solve the following by quadratic formula 1. 10x 19 0 b x b 4ac a x. x 1x x 8 0 x 4. x 7x x 14 0 x 6. 3x 18x x 3 0 x 8. 3x 1x 5 0 M. Winking Unit 3-4 page 85

15 Applications 1. Find the value of x that would make the diagram below accurate. x+ x x+4. A golf ball is hit with an initial vertical velocity of 80 fps h 16t 80t a. How high is the ball after seconds? b. How many seconds would it take the ball to hit the ground (the height would be h=0)? c. When will the ball reach 48 feet? d. What is the average rate of the change in height (i.e. vertical velocity) from t = 1 to t = seconds? (Hint: Find the slope between the points (1, ) and (, ). Evaluate the heights at t=1 and t= to find the corresponding coordinate) e. For what values of t is the domain appropriate? M. Winking Unit 3-4 page 86

16 3. A baseball his hit with an initial vertical velocity of 11 fps an the ball was struck 1 foot above ground. a. How high is the ball after seconds? h 16t 11t 1 b. How many seconds would it take the ball to hit the ground (the height would be h = 0)? c. When will the ball reach 30 feet? d. When does the ball reach a maximum height? e. What is the average vertical velocity from t = 0. to t = 0.9 seconds? 4. A person at a framing store is making a frame mat to go around a picture. The mat is a uniform inches around on each side. The picture s width is 5 less than twice the picture s height. The entire area of the frame with picture included is 1 square inches. What are the dimensions of the picture? M. Winking Unit 3-4 page 87

17 (REVIEW Unit -1) y 0.x Name: Consider the following EQUATIONS, make a table, plot the points, and graph what you think the graph looks like. y x 1. x y. y x x y 3. x y y 5x x y 5. What happens to the graph as the number in front of x gets larger? Smaller? 6. y x 7. x y y 0.5x x y 8. y x 1 x y y x x y 10. What happens to the graph as the number in front of x is negative? 11. What happens when you add a number or subtract a number from x? M. Winking M. Winking Unit 3-5 page 88

18 1. y x y x 3 y x 4 x y 13. x y 14. x y y x 3 x y 16. What happens when you add a number or subtract a number from x inside the parenthesis? 17.What is a possible equation for the following graphs. y = (x ) y = (x ) _ y = (x ) y = (x ) _ M. Winking Unit 3-5 page 89

19 18. Rewrite each of the following quadratics in vertex form by completing the square and graph. a. y x 4x 1 b. y x 6x 3 Domain: Domain: Range: Range: Vertex: (Max or Min) Vertex: (Max or Min) Axis of Symmetry: Axis of Symmetry: c. y x 1x 13 d. y x x 16 7 Domain: Domain: Range: Range: Vertex: (Max or Min) Vertex: (Max or Min) Axis of Symmetry: Axis of Symmetry: M. Winking Unit 3-5 page 90

20 (18 continued) Rewrite each of the following quadratics in vertex form by completing the square and graph. e. y x 7x 6 f. y x 11x 4 Domain: Domain: Range: Range: Vertex: (Max or Min) Vertex: (Max or Min) Axis of Symmetry: Axis of Symmetry: g. y x x h. y x x 15 Domain: Domain: Range: Range: Vertex: (Max or Min) Vertex: (Max or Min) Axis of Symmetry: Axis of Symmetry: M. Winking Unit 3-5 page 91

21 Name: f x x x 6 1. Consider the function: a. What are the zeros of the function (using factoring)? b. What is the axis of symmetry of the parabola? c. What is the vertex of the parabola (graph the parabola)? f x x 8x 15. Consider the function: a. What are the zeros of the function (using factoring)? b. What is the axis of symmetry of the parabola? c. What is the vertex of the parabola (graph the parabola)? f x x x 3 3. Consider the function: a. What are the zeros of the function (using factoring)? b. What is the axis of symmetry of the parabola? c. What is the vertex of the parabola (graph the parabola)? M. Winking Unit 3-6 page 9

22 3. The expression P x 70x 600 represents a company s profit for selling x items. a. What are the break-even point(s) for selling x items (i.e. how many items sold yields a profit of 0?) b. Is the vertex a minimum or maximum? c. If the model is accurate, how many items should the company sell to maximize their profit and what is the maximum profit? 4. The expression h 16t 400t 5 represents the height of a cannonball t seconds after it was fired. What is the maximum height of the cannon ball and how many seconds did it take to reach its maximum height? 5. The expression C x 44x 490 represents the cost in $1000 of dollars per year that company must spend out of pocket on each employee for health insurance for x number of employees. How many employees should the company hire to minimize their cost of health insurance? M. Winking Unit 3-6 page 93

23 Name: 1. Solve the following quadratic inequalities in one variable: a. x 3x 10 0 b. x 1x 1 0 c. x x 4 d. x 14x 0 e. 3x x 8 0 f. x 13x 4 0 M. Winking Unit 3-7 page 94

24 Name: 1. Given the formula for the perimeter of a rectangle is: l + w = P rewrite the formula so that it has been solved for the variable w.. Given the formula for Power in Watts of an electrical circuit is: P = I V where I is the resistance in Ohms and V is the voltage in volts, rewrite the formula so that it has been solved for V. 3. Given the formula for Centripetal Acceleration can be described by the formula: a = v r where v is the velocity in meters per second (m/s) and r is the length of the radius in meters. Assuming all variables represent positive values, rewrite the formula so that it has been solved for v. 4. Given the formula for the area of a trapezoid is: h (b 1 + b ) = P Rewrite the formula so that it has been solved for the variable b 1. M. Winking Unit 3-8 page 95

25 5. Given the function, f(x) = x + x, determine the average rate of change from x = 1 to x =. 6. Given the function, p(x) = x + 1, determine the average rate of change from x = 0 to x =. 7. Given the table of values for h(x), x h(x) What is the average rate of change from x = to x = 6? 8. Given the table of values for g(x), x g(x) What is the average rate of change from x = to x = 4? 9. Given the graph of q(x), 10. Given the graph of q(x), what is the average rate of change from x = 1 to x = 3. what is the average rate of change from x = 1 to x = 3. M. Winking Unit 3-8 page 96

26 11. A group of students visited Stone Mountain. They decided to walk up to the top of the mountain. At 3:00 pm they started walking and according to their GPS when they were at the bottom of the mountain their elevation was 861 feet above sea level. At 3:45 pm they were at the top of the mountain which was 1686 feet above sea level. What is the students average rate of change in feet per minute? 1. A college student is driving from Athens to Panama City Beach for a vacation. The student left Athens at 1:00pm and arrived at the beach at 5:15pm but gained an hour due to the Standard Time Zone change. The trip was exactly 350miles. What was the student s average rate of speed in miles per hour? Do you think the student ever traveled more than 58 miles per hour? M. Winking Unit 3-8 page 97

27 Name: 1. To rent a bicycle on the beach board walk, the initial cost is $10 plus an additional charge of $5 per hour. Create a linear function model and graph.. Let h(x) be the number of person-hours it takes to assemble x engines in a factory. The company s accountant determines that the time it takes depends on start-up time and the number of engines to be completed. It takes 6.5 hours to set up the machinery to make the engines and about 5.5 hours to completely assemble one. Create the function h(x). 3. Two students were selling cookies for a fundraiser. They both started on the first day of the month. The graph shows how many boxes Cindy sold in total after each day. If Veronica hadn t sold any boxes by the 3 rd day of the month but then started selling cookies at the exact same rate, how many cookie boxes will Veronica have sold by the 1 th day of the month? 4. A babysitter charges parents $10 for coming to their house and then $6 for every hour she babysits. Create function c(x), where x = the number of hours and c(x) represents the amount the babysitter charges. M. Winking Unit 3-9 page 98

28 5. The function graphed shows the decibel level a car horn makes at varying distances away from the horn. a. At prolonged period of listening anything above 90 decibels can cause permanent ear damage. How far must you be away from the horn to prevent potential ear damage? b. How many decibels would the car horn be at 70 feet away? 6. Bonnie is renting space at a local private school to offer math tutoring. The school charges her a monthly fee to use classrooms after school and she charges the students a flat rate per hour. She models her profit using the function: p(x) = c x k. Explain what you think the variable x and the parameters c and k should each represent. 7. A pair of shoes cost $135 and has been discounted 38%. What is the sale price of the shoes? 8. A person paid their electricity bill late. The bill is $78. The electric company charges an 1% late charge. What is the total cost of the bill after paying it late? 9. If a store has marked down all of its products 0%, create a function that would represent the new cost after the discount, d(x), where x is the original price of the item before the discount. 10. If a furniture store buys furniture at wholesale cost of x and marks up the furniture by 95% to sell on their showroom floor, create a function p(x) that represents the price of the item in the store. 11. A manager at a jewelry store buys jewelry at whole sale cost of x. To set a price on the item in the store she first adds $5 and she calls this function a(x). Then, she increases by 80% and calls this combined price increase p(a(x)). Create a function model that expresses the overall increase in price. M. Winking Unit 3-9 page 99

29 1. Describe each of the following as examples of relationships that could be modeled by: LINEAER functions, QUADRATIC functions, EXPONENTIAL functions, or NONE OF THESE. M. Winking Unit 3-9 page 100

30 13. Describe each of the following as examples of relationships that could be modeled by: LINEAER functions, QUADRATIC functions, EXPONENTIAL functions, or NONE OF THESE. The Sequence: {4, 1, 36, 108, 34, } The Sequence: {3, 9, 15, 1, 7, 33, } M. Winking Unit 3-9 page 101