Algebra 2 Notes Section 4.1: Graph Quadratic Functions in Standard Form Objective(s): Vocabulary: I. Quadratic Function: II. Standard Form: III. Parabola: IV. Parent Function for Quadratic Functions: Vertex Axis of symmetry V. Vertex of a parabola: The x-coordinate of the vertex: VI. Axis of symmetry: A line that divides The equation of the axis of symmetry: VII. Properties of the Graph of y = ax 2 + bx + c: (0,c) x = b 2a x = b 2a
Notes 4.1 page 2 VIII. Minimum value of a quadratic function (see Glossary): IX. Maximum value of a quadratic function (see Glossary): Examples: 1 2 1. Graph y x. Compare the graph with the graph of y = x 2. 2 2. Graph y = 2x 2 + 4. Compare the graph with the graph of y = x 2.
Notes 4.1 page 3 3. Graph y = x 2 + 6x 8. 4. Tell whether the function y = 2x 2 + 4x + 3 has a minimum value or a maximum value. Then find the minimum or maximum value. 5. A video store sells about 150 DVDs a week at a price of $20 each. The owner estimates that for each $1 decrease in price, about 25 more DVDs will be sold each week. How can the owner maximize weekly revenue? (Hint: Revenue = Price Number of DVDs sold)
Algebra 2 Notes Section 4.2: Graph Quadratic Functions in Vertex or Intercept Form Objective(s): Vocabulary: I. Vertex Form: II. Characteristics of the graph of y = a(x h) 2 + k: The graph of y = a(x h) 2 + k is the III. Intercept Form: IV. Graph of Intercept Form y = a(x p)(x q): y = a(x + p)(x + q) V. FOIL Method: Examples: 1. Graph y = ½ (x 3) 2 5.
Notes 4.2 page 2 2. The Tacoma Narrows Bridge in Washington has two towers that each rise 307 feet above the roadway and are connected by suspension cables. 1 Each cable can be modeled by the function y x 2 1400 27. What 7000 are the minimum and maximum distances between the suspension cables and the roadway? 3. Graph y = x(x 4). 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) = 16t(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? 5. Write y = 3(x 4)(x + 6) in standard form. 6. Write f(x) = ½ (x + 8) 2 + 35 in standard form.
Algebra 2 Notes Section 4.3: Solve x 2 + bx + c = 0 by Factoring Objective(s): Vocabulary: I. Monomial: II. Binomial: III. Trinomial: IV. Special Factoring Patterns: Pattern Name Pattern Example V. Standard form of a Quadratic equation: (in one variable) VI. Roots: VII. Zero Product Property: VIII. Zeros of a Function: Examples: 1. Factor the expression. a. x 2 + 14x + 48 b. x 2 9x + 20 c. x 2 121 d. x 2 + 14x + 49 e. x 2 24x + 144
Notes 4.3 page 2 2. Find the roots of the equation x 2 x 42 = 0. 3. You have a rectangular vegetable garden in your backyard that measures 15 x 10. 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. 4. Find the zeros of the function by rewriting the functions in intercept form. a. y = x 2 + 3x 28 b. y = x 2 4x + 4
Algebra 2 Notes Section 4.4: Solve ax 2 + bx + c = 0 by Factoring Objective(s): Vocabulary: I. Factoring ax 2 + bx + c when a 1: ax 2 + bx + c = ( ) ( ) = Find integers k, l, m, and n such that k l m n must be must be c > 0 m and n have c < 0 m and n have II. Factoring ax 2 + bx + c when a 1: (This method is commonly referred to as factoring by grouping ) example: Factor 3x 2 + 14x 5 Step 1: Find two numbers whose product is: and whose sum is: a c b Step 2: Rewrite the middle term, 14x, using the two numbers you found in step 1. You will have a polynomial with four terms. Step 3: Group the first two terms and factor; group the last two terms and factor. There should be an common binomial factor in each of these. Factor the common binomial from each term. III. Factoring out monomials: IV. Solving quadratic equations: If the left side of the quadratic equation
Notes 4.4 page 2 V. Factoring and zeros: To find the The maximum or minimum occurs Examples: 1. Factor. a. 3x 2 10x + 8 b. 6x 2 + x 15 2. Factor. a. 81x 2 25 b. 49x 2 + 112x + 64 c. 9x 2 66x + 121
3. Factor. Notes 4.4 page 3 a. 3x 2 300 b. 8x 2 + 28x 120 c. 7x 2 63x d. 25x 2 + 60x 35 4. Solve the equation. a. 4x 2 17x 15 = 0 b. 3x 2 + 22x + 60 = 14x 48 5. You are designing a garden for the grounds of Adams 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 22 by 15, and your budget will allow for enough stone to cover 120 square feet. What should be the width of the border?
Algebra 2 Notes Section 4.5: Solve Quadratic Equations by Finding Square Roots Objective(s): Vocabulary: I. Square root: II. Principal square root: III. Radical: IV. Radicand: V. Properties of Square Roots (a > 0, b > 0): Product Property: Quotient Property: VI. Simplifying square roots: A square-root expression is simplified if: VII. Rationalizing the denominator: (see Glossary) VIII. Conjugates: Their product is IX. Modeling dropped objects: Use the formula where h = t = h o =
Examples: Notes 4.5 page 2 1. Simplify the expression. a. 75 b. 7 35 c. 100 169 d. 11 144 2. Simplify the expression. a. 2 15 b. 3 7 2 c. 4 5 2 3. Solve. 1 2 3 a. 2x 2 15 = 65 b. Solve x 4 11 4. If you drop an object off the roof of an apartment building that is 240 tall, about how long will it take the object to hit the ground?
Algebra 2 Notes Section 4.6: Perform Operations with Complex Numbers Objective(s): Vocabulary: I. Imaginary unit, i: Note that i 2 = The imaginary unit i can be used II. The Square Root of a Negative 1. Number (Properties): 2. III. Complex number in standard form: Real part: Imaginary part: Complex Numbers (a + bi) Real Imaginary Numbers Numbers (a + 0i) (a + bi, b 0) IV. Imaginary number: a + bi where 1 5 2 2 + 3i 5 5i V. Pure imaginary number: a + bi where π 2 Pure Imaginary Numbers (0 + bi, b 0) 4i 6i VI. Sum of complex numbers: VII. Difference of complex numbers: VIII. Multiplying complex numbers: IX. Complex conjugates: Read the application of complex numbers at the top of page 277 (Example #3).
Examples: Notes 4.6 page 2 1. Solve 2x 2 + 18 = 72. 2. Write the expression as a complex number in standard form. a. (12 11i) + ( 8 + 3i) b. (15 9i) (24 9i) c. 35 (13 + 4i) + i d. 5i(8 9i) e. ( 8 + 2i)(4 7i) 3. Write the quotient 3 4i 5 i in standard form. 4. Each component of the circuit has been labeled with its resistance or reactance. Find the impedance of the circuit. Use the fact that the impedance for a series circuit is the sum of the impedances for the individual components. Component and Symbol Resistance or reactance Resistor Inductor Capacitor R L C Impedance R Li Ci
Algebra 2 Notes Section 4.7: Complete the Square Objective(s): Vocabulary: I. Perfect square trinomial (p. 253): II. Completing the square: (see Glossary) The method of completing the square can be used to To write a quadratic function in vertex form, Examples: 1. Find the value of c that makes x 2 26x + c a perfect square trinomial. Then write the expression as the square of a binomial. 2. Rewrite in vertex form by completing the square. Then identify the vertex of each parabola. a. y = x 2 10x + 1 b. y = x 2 + 18x + 95 c. y = 3x 2 36x + 150 d. y 14 = 2x 2 + 8x
Notes 4.7 page 2 3. The height y (in feet) of a baseball t seconds after it is hit is given by the function y = 16t 2 + 96t + 3. Use the method of completing the square to find the maximum height of the baseball. 4. 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 2 + 64t + 50. Find the maximum height of the ball.
Algebra 2 Notes Section 4.8: Use the Quadratic Formula and the Discriminant Objective(s): Vocabulary: I. The Quadratic Formula: II. Discriminant: III. Using the Discriminant of ax 2 + bx + c = 0 Value of discriminant Number and type of solutions y y y Graph of y = ax 2 + bx + c x x x Number of x-intercepts IV. Modeling Launched Objects: Object is dropped: Object is launched or thrown: h = t = v o = h o =
Examples: Notes 4.8 page 2 For #1-3, solve. 1. x 2 5x = 7 2. x 2 6x + 10 = 0 3. 16x 2 23x = 17x 25 4. Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. a. x 2 + 10x + 23 = 0 b. x 2 + 10x + 25 = 0 c. x 2 + 10x + 27 = 0 5. 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 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?
Algebra 2 Notes Section 4.10: Write Quadratic Functions and Models Objective(s): Vocabulary: I. Best-fitting quadratic model (see Glossary): Examples: 1. Write a quadratic function for the parabola shown. 2. Write a quadratic function for the parabola shown. 3. Use a graphing calculator to write a quadratic function in standard form for the parabola that passes through the points ( 2, 30), (1, 6), and (4, 36). 4. The drama club at a high school sells T-shirts as a fundraiser. The table shows data from the last four years for the price charged for a T-shirt, x, and the total revenue earned from selling them, y. Use a graphing calculator to find the best-fitting quadratic model for the data. x 8 10 12 14 y 1180 1450 1675 1550