CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING

Transcription

1 CHAPTER 1 QUADRATIC FUNCTIONS AND FACTORING Big IDEAS: 1) Graphing and writing quadratic functions in several forms ) Solving quadratic equations using a variety of methods 3) Performing operations with square roots and complex numbers Section: Essential Question 1 1 Graph Quadratic FUNctions in Standard Form How do you graph quadratic functions in standard form? When exploring the graph of quadratic equations it is helpful to understand the PARENT FUNCTION y x. Complete the table and graph the parent function in the quadratic equation family. x y Key Vocabulary: Quadratic Function Parabola Vertex Axis of Symmetry Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #1

2 Partner Activity: A) Write and graph quadratic functions with the described a values. B) Compare your graph with the graph of y x. C) Draw a conclusion(s) based on the a value. 1. a 0 Equation: Comparison: Conclusion:. a 0 Equation: Comparison: Conclusion: 3. a 1 Equation: Comparison: Conclusion: 4. a 1 Equation: Comparison: Conclusion: Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #

3 Properties of the Graph y ax bx c If (the a value is ), then the graph opens. If (the a value is ), then the vertex is a. If, then the graph is than y x. If (the a value is ), then the graph opens. If (the a value is ), then the vertex is a. If, then the graph is than y x. The axis of symmetry is AND the vertex has an x-coordinate, so are the coordinates for the vertex. The is c, so the point is always on the graph of the parabola. Ex 3) Graph y x x 6 8. Label the vertex and axis of symmetry. Vertex: Axis of Symmetry: Y intercept: Use the axis of symmetry to plot additional points. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #3

4 Ex 4) Tell whether the function y x 4x 3 has a maximum or minimum value. Find the maximum or minimum value. a value is negative Ex 5) A video store sells about 150 DVDs a week at a price of $0 each. The owner estimates that for each $1 decrease in price, about 5 more DVDs will be sold each week. How can the owner maximize weekly revenue? Note: Revenue = Price Quantity Closure: How is the vertex of a parabola related to its axis of symmetry? The x-coordinate of the vertex yields the equation for the axis of symmetry Why is it useful to know the axis of symmetry when graphing a parabola? Can a quadratic function have a maximum AND a minimum value? Why or why not? No. The maximum/minimum of a quadratic function occurs at the vertex. A parabola has only one vertex. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #4

5 Section: Essential Question 1 Graph Quadratic FUNctions in Vertex or Intercept Form How do you graph quadratic functions in vertex form? How do you graph quadratic functions in intercept form? Key Vocabulary: Quadratic equation of the form: Vertex Form Vertex has coordinates: Axis of symmetry is: Quadratic equation of the form: Intercept Form x-intercepts are located at and The vertex is located between the x-intercepts and can be found using T h e F O I L M e t h o d Used to multiply two binomials (two term expressions) together First Outer Inner Last x x 3 x 3x x 6 x x 6 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #5

6 1. Ex 1) Graph y x 3 5 Vertex: Axis of Symmetry: Ex ) The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables as shown. Each cable can be modeled by the function y 1 x1400 7, where x and 7000 y are measured in feet. What are the minimum and maximum distances between the suspension cables and the roadway? Ex 3) Graph y x x 4. x intercepts: Vertex: Axis of Symmetry: Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #6

7 Ex 4) If an object is propelled straight upward from Earth at an initial velocity of 80 feet per second, its height after t seconds is given by the function h( t) 16 t( t 5), where t is the time in seconds after the object is propelled and h is the object s height in feet. a. How many seconds after it is propelled will the object hit the ground? b. What is the object s maximum height? The maximum height is 100 feet. Ex 5) Write y 3 x 4 x 6 standard form. in ( ) in standard form. Ex 6) Write f x x Closure: When you see an equation of a quadratic function in vertex form, how do you know if the vertex is a maximum or a minimum point? When a quadratic function is in intercept form, how can you find the x-values of the intercepts AND the vertex? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #7

8 Section: 1 3 Solve x bx c 0 by Factoring Essential Question How can factoring be used to solve quadratic equations when a = 1? Key Vocabulary: Monomial A term expression Examples: 3, x, 5ab Binomial A term expression Examples: a b, 3x 4 Trinomial A term expression Example: x x 5 Root S p e c i a l P r o d u c t s Type Formula Example Difference of Squares a b x 16 Perfect Square Trinomial a ab b a ab b x x 8x16 10x5 If the product of two expressions is zero, then or of the expressions equal zero. Example: If Z e r o P r o d u c t P r o p e r t y If A and B are expressions and AB 0, then or. x x 3 0, then x 0 or x 3 0. Factor the expression. 1) x 14x 48 ) x 9x 5 3) x 9 4) q 100 5) y 16y 64 6) w w ) m 11 8) r 14r 49 9) p 4 p 144 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #8

9 Ex 1) What are the roots of the equation x x 4 0? A. -7, 6 C. -6, 7 B. -7, -6 D. 6,7 Ex ) You have a rectangular vegetable garden in your backyard that measures 15 feet by 10 feet. You want to double the area of the garden by adding the same distance x to the length and width of the garden. Find the value of x and the new dimensions of the garden. Find the zeros of the function by rewriting the function in intercept form. Ex 3) y x 3x 8 Ex 4) y x 4x 4 Closure: How can you recognize when a trinomial of the form square trinomial? x bx c 0 is a perfect If an equation of the form x bx c 0 has only one root, what kind of trinomial is x bx c? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #9

10 Section: 1 4 Solve ax bx c 0 by Factoring Essential Question How can factoring be used to solve quadratic equations when a 1? Key Vocabulary: Greatest Common Factor ax bx c Factoring Helpful Hints ax bxc,, then both ax bx c ax bx c ax bx c,, then both,, one of each,, one of each Ex 1) Factor the expression. a. 3x 10x 8 b. 6x x 15 c. 81x 5 d. 49z 11z 64 e. 9r 66r 11 f. 3x 300 g. 8m 8m 10 h. 7t 63t i. 5y 60y 35 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #10

11 Ex 3) Solve the equation. a. 4 x x b. 3y y y c. 6 x x d. 7 x 7 x Ex 4) You are designing a garden for the grounds of you high school. You want the garden to be made up of a rectangular flower bed surrounded by a border of uniform width to be covered with decorative stones. You have decided that the flower bed will be feet by 15 feet, and your budget will allow for enough stone to cover 10 square feet. What should be the width of the border? Ex 5) An internet service provider sells high-speed internet service for $30 per month to 1500 customers. For each $1 increase in price, the number of customers will decrease by 5. How much should the company charge in order to maximize monthly revenue? What is the maximum monthly revenue? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #11

12 Section: Essential Question 1 5 Solve Quadratic Equations by Finding Square Roots How can you use square roots to solve a quadratic equation? Discovering the Concept: Solve x 5 by factoring. A square root of a number is the solution of the equation. Therefore the equation x b can be rewritten in the form ; factored into the form ; and solved to show that each square root has a positive and negative solution, i.e.. Since the square of a real number is never negative, the equation has no real-number solution if ( ). Key Vocabulary: Square Root Principal Square Root Radical If, then. The square root of a number. An expression of the form or. Radicand Index Expression the radical sign. Number that represents the value. 3 3x Simplified Radical Conjugate No radicand has a - factor other than 1. There is no radical in the. Binomials that are the and of the same two terms. Used to the denominator when simplifying radical expressions. Examples: a b and a b x 3 and x 3 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #1

13 P r o p e r t i e s o f R a d i c a l s: Product Property ab The square root of a product is the product of the square roots. Division Property a b The square root of a quotient is the quotient of the square roots. Simplify the expression. Ex 1) 75 Ex ) 7 35 Ex 3) Ex 4) Ex 5) 15 Ex 6) 4 5 Solve. Ex 7) 1 x 1 65 Ex 8) x 5 Ex 9) When an object is dropped, its height h in feet above the ground after t seconds can be modeled by the function h 16t h0 where h 0 is the object s initial height in feet. If you drop an object off the roof of an apartment building that is 40 feet tall, about how long will it take the object to hit the ground? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #13

14 Section: Essential Question 1 6 Perform Operations with Complex Numbers How do you perform operations with complex numbers? Consider the equation x 1 it has no real solution we cannot take the square root of a negative number until today. Mathematicians invented the imaginary number i to solve this equation. The imaginary number i has many applications in electricity one example is when an electrical current flows through a circuit with a resistor, and inductor and a capacitor in defining electromotive force. i also plays an important role in fractal geometry. Key Vocabulary: Imaginary Unit, i Complex Number Standard Form of a Complex Number Imaginary Number Pure Imaginary Number Complex Conjugates Complex Plane i and i Examples: 3 i 3 and 5 i 5, where a and b are real numbers and i is the imaginary unit. A complex number a bi, where b 0. A complex number a bi, where a 0 and b 0. Examples:,, A pair of complex numbers whose middle signs are. The of two complex conjugates will always be a. Example: x 3i and x 3i A coordinate plane in which each coordinate ( abrepresents, ) a complex number a bi. The axis (x-axis) is the axis. The axis (y-axis) is the axis. Absolute Value of a Complex Number The absolute value of a complex number z a bi, denoted z, is a nonnegative real number defined as. Represents the between in the complex plane. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #14

15 Sum/Difference of Complex Numbers add/subtract the parts and add/subtract the parts a bi c di Product of Complex Numbers FOIL a bi c di Hint: When simplifying radicals (even sums, differences, products and quotients) with negative always take care of the first! Caution: The rule a b ab does not apply when a and b are NEGATIVE! You must express each radical as pure imaginary numbers before simplifying! Simplify. Ex 1) Ex ) 75 5 Ex 3) Solve. Ex 4) x 18 7 Ex 5) 4x 5 3 Write the expression as a complex number in standard form. 1 11i 8 3i 15 9i 4 9i Ex 6) Ex 7) Ex 8) i i Ex 9) 5i8 9i Ex 10) 8 i4 7i Ex 11) 3 4 i 5 i Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #15

16 Ex 1) Impedance R Li Ci Ex 13) Plot each complex number in the same complex plane. a. 4 i imaginary b. 1 3i c. 4i real d. i Ex 14) Find the absolute value the expression. a. 5 1i b. 17i Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #16

17 Section: Essential Question 1 7 Complete the Square How is the process of completing the square used to solve quadratic equations? Key Vocabulary: Completing the Square The process of to a quadratic expression of the form x bx to make it a. 1. Transform the equation so that the C o m p l e t i n g t h e S q u a r e Steps: Example: c is alone on the right side.. If a is not equal to 1, then both sides of the equation by. 3. the square of half the coefficient of the first degree term to both sides. Step 1: Step : Step 3: 4. the left side. Step 4: 5. by taking the square root of both sides of the equation. Step 5: x x 8x 14 0 x 8x 1 x 8x 14 4 x 4x x 7 x 4x x 3 x 3 x i 3 xi 3 Solve the equation by finding square roots. Ex 1) x 0x Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #17

18 Ex ) Find the value of c that makes x 6x c a perfect square trinomial, then write the expression as the square of a binomial. Solve by completing the square. Ex 3) x 10x 1 0 Ex 4) 3 x 5 x Ex 5) 7x 1x 3 0 Ex 6) The area of the triangle shown is 144 square units. What is the value of x? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #18

19 Ex 7) Write y x x in vertex form. Identify the vertex. Ex 8) The height y (in feet) of a ball that was thrown up in the air from the roof of a building after t seconds is given by the function y 16t 64t 50. Find the maximum height of the ball. Closure: If x bx c 0 is a perfect square trinomial and b is an odd integer, what do you know about the value of c? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #19

20 Section: Essential Question 1 8 Use the Quadratic Formula and the Discriminant How do you use the Quadratic Formula to solve quadratic equations? How do you use the disciminant? Quadratic equations are used in many, many applications when our factoring techniques do not apply we can Complete the Square or use the formula derived from Completing the Square called the. Every quadratic equation can be solved by using the Quadratic Formula! Use the Completing the Square Method to derive the Quadratic Formula: x ax bx c ax bx c 0 0 a a a a b c x x 0 a a b c x x a a b c x a a 1 b a Key Vocabulary: Let a, b, and c be real numbers such that a 0, then Quadratic Formula can be used to solve in standard form ax bx c 0. Discriminant Radicand of the quadratic formula:. Used to determine the nature of the roots/solutions. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #0

21 U s i n g t h e D i s c r i m i n a n t Value of the Discriminant: Number of Roots: Graphical Representation: Use the quadratic formula to solve. Ex 1) x 5x 7 Ex ) x 6x 10 0 Ex 3) 16x x 3 17x 5 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #1

22 Find the discriminant of the quadratic equation. Use the discriminant to determine the number of solutions for the equation. Ex 4) x 10x 3 0 Ex 5) x 10x 5 0 Ex 6) x 10x 7 0 Ex 7) For an object that is launched or thrown the equation, where v 0 h 16t v0t h0 stands for an initial velocity and h 0 stands for the initial height is used to model the height of the object as a function of time. A basketball player passes the ball to a teammate. The ball leaves the player s hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long is the ball in the air? Closure: Does the discriminant give the solution of a quadratic equation? How can you tell if a quadratic equation has one or two solutions? How are the solutions of a quadratic formula related if the discriminant is negative? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #

23 Section: Essential Question 1 9 Graph and Solve Quadratic Inequalities How do you solve quadratic inequalities in two variables? How do you solve quadratic inequalities in one variable? y ax bxc Graphing a Quadratic Inequality in Two Variables y ax bxc y ax bx c 1. the parabola with the equation. Use a curve for and. Use a curve for and. y ax bx c. a inside the parabola to determine whether that point is in the solution of the inequality. 3. the parabola if the test point is a solution. the parabola if the test point is not a solution. Ex 1) Graph y x 4x Ex ) A computer desk with a solid glass top can safely support a weight W (in pounds) provided W 50x, where x is the thickness of the desktop (in inches). Graph the inequality. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #3

24 Ex 3) Graph the system of inequalities. y x 3 y x 4x ax bx c Solving a Quadratic Inequality in One Variable Algebraically 0 ax bx c 0 ax bx c 0 ax bx c 0 1. the equivalent equation ax bx c 0. the solutions as on a number line/coordinate plane. Use for and. Use for and. 3. Use the critical values to divide the number line/coordinate plane into partitions. Choose a from each partition. 4. If the test point is a solution then, the interval. If the test point is not a solution, then shade the interval. Solve algebraically. Ex 4) 3x 9x 1 0 Ex 5) x 7x 4 Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #4

25 Ex 6) Solve by graphing. x x Ex 7) Solve x x 8 by using a table. x y Ex 8) Solve x 6x 9 0. Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #5

26 Ex 9) Solve x 3x 5 0. Ex 10) Solve 1 x 0. x 6 Closure: What is the difference between solving a quadratic inequality in one variable and solving a quadratic inequality in two variables? Student Notes Honors Algebra II Chapter 1 Quadratic Functions and Factoring Page #6