Chapter 5: Quadratic Function. What you ll learn. Chapter 5.1: Graphing Quadratic Functions. What you should learn. Graphing a Quadratic Function:

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1 Chapter 5: Quadratic Function What you ll learn Chapter 5.1: Graphing Quadratic Functions What you should learn Graphing a Quadratic Function: Graphing Calculator activity, P249 The graph of a Quadratic Equation: Example: Graph y = x + x + 12 y = x x 6

2 Additional Forms of a Quadratic Function: Example: Graph y = 2(x 1) + 3 y = (x + 5) + 2 y = 2(x 3)(x + 1) y = 2x(x 4) Example: Write the quadratic function in standard form. y = (x 6)(x 4) y = 4(x 7) + 2 y = (x + 2) 3 y = 3(x + 1)(x 5)

3 Example: Suppose that a group of high school students conducted an experiment to determine the number of hours of study that leads to the highest score on a comprehensive year-end exam. The exam score y for each student who studied for x hours can be modeled by y = x x Which amount of studying produced the highest score on the exam? What is the highest percent score the model predicts? Example: The path of a ball thrown by a baseball player forms a parabola with equation: y = (x 49) + 8.5, where x is the horizontal distance in feet of the ball from the player and y is the height in feet of the ball. How far does the ball travel before it again reaches the same height from which it was thrown? How high was the ball at its highest point? Chapter 5.2: Solving Quadratic Equations by Factoring What you should learn Factoring Quadratic Expressions Example: Factor the following: x 2x 48 x 21x + 80 x + 5x + 6 x 3x 4 x + 9x 36 x 10x + 21

4 Example: Factor the following: 4y 4y 3 5x + 17x + 14 Difference of Two Squares: Perfect Square Trinomial: GCF: Example: Factor the following: 16y x + 3x + 3 4z 12z + 9 4u 36 36w + 60w x 9 14x + 2x 12 16x + 8x + 1 3v 18v 30u 57u + 21

5 Solving Quadratic Equations by Factoring: Example: Solve by factoring: 9t 12t + 4 = 0 3x 6 = x 10 2w 10w = 23w w Example: A painter is making a rectangular canvas for her next painting. She wants the length of the canvas to be 4 ft more than twice the width of the canvas. The area of the canvas must be 30 ft 2. What should the dimensions of the canvas be? Find Zeros of Quadratic Functions: Example Find the zeros of y = 3x + 14x 5 Find the zeros of y = x + 8x + 15

6 Example: You own an amusement park that averages 75,000 visitors per year who each pay a $12 admission charge. You plan to lower the admission price to attract new customers. It has been shown that each $1 decrease in price results in 15,000 new visitors. What admission should you charge to maximize your annual revenue? What is the maximum revenue? Chapter 5.3: Solving Quadratic Equations by Finding Square Roots Goals: Square root: Radicand: Radical: Investigating Properties of Square Roots P264 Properties of Square Roots (a > 0, b > 0) Product Property: Quotient Property: A square-root expression is considered simplified if: Example: Simplify the expression Rationalizing the Denominator:

7 Example: Simplify the expression Solving Quadratic Equations: Example: Solve the following equations. 3 5x = 9 4x 6 = 42 3(x 2) = (x 4) = 6 Using Quadratic Models in Real-Life: Example: The tallest building in the United States is in Chicago, Ill. It is 1450 ft. tall. How long would it take a penny to drop from the top of this building? How fast would the penny be traveling when it hits the ground if the speed is given by s = 32t where t is the number of seconds since the penny was dropped? Chapter 5.4: Complex Numbers Goals:

8 Imaginary Numbers: Example Solve 2x + 26 = 10 Solve (x + 1) = 5 Complex Numbers: Complex Plane Example: Plot the complex numbers in the complex plane. 4 i i To add or subtract complex numbers:

9 Example: Write the expression as a complex number in standard form. ( 1 + 2i) + (3 + 3i) (2 3i) (3 7i) 2i (3 + i) + (2 3i) Multiplying Complex Numbers: Example: Write the expression as a complex number in standard form. i(3 + i) (2 + 3i)( 6 2i) (1 + 2i)(1 2i) Dividing Complex Numbers: Example: Write the quotient in standard form. Example: Write the expressions as a complex number in standard form. 3i(9 i) ( 1 + 4i)(3 6i) i 1 2i

10 Chapter 5.5: Completing the Square Goals: Perfect Square Trinomial: Completing the Square Example: Find the value of c that makes x 3x + c a perfect square trinomial. Then write the expression as the square of a binomial. Example: Solve by completing the square. x 6x 8 = 0 5x 10x + 30 = 0 x + 4x 1 = 0 by 3x 12x + 16 = 0

11 Example: Under certain road conditions, the formula for a car s stopping distance is given by d = 0.1s + 1.1s. If a driver leaves 5 car lengths, approximately 75 ft, between him and the driver in front of him, what is the maximum speed he can drive and still stop safely? Example: You have 30 ft of chain link fence to make a rectangular enclosure for your dog. A pet store owner recommended that an enclosure for one dog be at least 48 ft 2 in area. What should the dimensions of the enclosure be to make the area 48 ft 2. Writing in Vertex Form Example: Write the quadratic function y = x + 6x + 16 in vertex form. What is the vertex of the function s graph? Example: Write the quadratic function y = x + 3x + 3 in vertex form. What is the vertex of the function s graph? Example: An agricultural researcher finds that the height h (in inches) of one type of pepper plant can be modeled by the function h = 0.88r + 8.8r + 20 where r is the amount of rainfall (in inches) that fell during the growing season. How much rain would maximize the height of the pepper plants? What is the maximum height?

12 Chapter 5.6: The Quadratic Formula and the Discriminant Goals The Quadratic Formula Example: Solve using the quadratic formula. 3x + 8x = 35 2x + x = x 2x x 5 = 2x + 13 x = 2x 5 2x = 2x + 3 The Discriminant: Example: Find the discriminant of the quadratic equations and give the number and types of solutions of the equations. 9x + 6x + 1 = 0 9x + 6x 4 = 0 9x + 6x + 5 = 0

13 Example: The water in a large fountain leaves the spout with a vertical velocity of 30 ft per second. After going up in the air it lands in a basin 6 ft below the spout. If the spout is 10 ft above the ground, how long does it take a single drop of water to travel from the spout to the basin? Use the model h = 16t + v t + h. Chapter 5.7: Graphing and Solving Quadratic Inequalities Goals: Graphing a Quadratic Inequality: Example: Graph y 2x 5x 3 Example: You are making a photo album. Each album page needs to be able to hold 6 square pictures. If the length of one side of each picture is x, then A 6x is the area of one album page. Graph this function. If you have an album page that has an area of 70 square inches, will it be able to accommodate 6 pictures with 3-inch sides?

14 Systems of Quadratic Inequalities: Example: Graph the system of inequalities y x + 9 y x + 5x 6 Solving Quadratic Inequalities in One Variable: Example: Solve x 5x Example: Solve x 11x Example: Solve x 9x + 36 > 0. Example: Solve 3x + x + 7 < 0.

15 Solving Quadratic Inequalities in One Variable: Example: Solve 2x x > 3 Example: Solve 3x + 11x 4 Example: Suppose a study was conducted to test the average reading comprehension of a person x years of age. The study found that the number of points P(x) scored on a reading comprehension test could be modeled by: P(x) = 0.017x + 1.9x + 31, 5 x 95. At what ages does the average person score greater than 60 points on this test? Chapter 5.8: Modeling with Quadratic Functions Goals Example: Write a quadratic function for the parabola shown.

16 Example: Write a quadratic function for the parabola shown. Writing a Quadratic in Standard Form Example: A group of students dropped a rubber ball and measured the height in inches of the ball for each of its successive bounces. The results are shown in the table. Find the quadratic model in standard form for the data using the first three points. Best-Fitting Quadratic Model- Quadratic Regression: On graphing calculator: Example: A bank adjusts its interest rates for new certificates of deposits daily. The table shows the interest rates on the first of the month for January through May. Find the best-fitting quadratic model for the data. According to the model, during which month did the certificates of deposit have the highest interest rate. What was that rate? Example: Find the best-fitting quadratic model for the data in the table.