Quadratic Functions and Equations


 Roxanne Dana Ray
 11 months ago
 Views:
Transcription
1 Quadratic Functions and Equations
2 Quadratic Graphs and Their Properties Objective: To graph quadratic functions of the form y = ax 2 and y = ax 2 + c.
3 Objectives I can identify a vertex. I can grapy y = ax 2. I can compare widths of parabolas. I can graph y = ax 2 + c. I can use the falling object model.
4 Vocabulary A quadratic function is a type of nonlinear function that models certain situations where the rate of change is not constant. The graph of a quadratic function is a symmetric curve with the highest or lowest point corresponding to a maximum or minimum value. Standard Form of a Quadratic Function: A quadratic function is a function that can be written in the form y = ax 2 + bx + c, where a 0. This form is called the standard form of a quadratic function. Examples: y = 3x 2 y = x y = x 2 x 2
5 Vocabulary The simplest quadratic function f x = x 2 or y = x 2 is the quadratic parent function. The graph of a quadratic function is a Ushaped curve called a parabola. You can fold a parabola so that the two sides match exactly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry.
6 Vocabulary The highest or lowest point of a parabola is its vertex, what is on the axis of symmetry. If a > 0 in y = ax 2 + bx + c, the parabola opens upward. The vertex is the minimum point, or lowest point, of the parabola. If a < 0 in y = ax 2 + bx + c, the parabola opens downward. The vertex is the maximum point, or highest point, of the parabola.
7 Identifying a Vertex
8 Practice Identify the vertex.
9 Practice
10 Vocabulary You can use the fact that a parabola is symmetric to graph it quickly. First, find the coordinates of the vertex and several points on one side of the vertex. Then reflect the points across the axis of symmetry. For graphs of functions of the form y = ax 2, the vertex is at the origin. The axis of symmetry is the y axis, or x = 0.
11 Graphing y = ax 2 Graph the function. Make a table of values. What are the domain and range? 1. y = 1 3 x2 2. y = 3x 2 3. y = 4x 2
12 Practice Graph each function. Then identify the domain and range of the function. 1. y = 4x 2 2. y = 1 3 x2 3. f x = 1.5x 2 4. f x = 2 3 x2
13 Vocabulary The coefficient of the x 2 term in a quadratic function affects the width of the parabola as well as the direction in which it opens. When m < n, the graph of y = mx 2 is wider than the graph of y = nx 2.
14 Comparing Widths of Parabolas Use your calculator to graph. What is the order, from widest to narrowest, of the graphs of the quadratic functions. f x = 4x 2, f x = 1 4 x2, f(x) = x 2 f x = x 2, f x = 3x 2, f x = 1 3 x2
15 Practice Order each group of quadratic functions from widest to narrowest graph. y = 3x 2, y = 2x 2, y = 4x 2 y = 1 2 x2, y = 5x 2, y = 1 4 x2 f x = 5x 2, f x = 3x 2, f x = x 2 f x = 2x 2, f x = 2 3 x2, f x = 4x 2
16 Vocabulary The y axis is the axis of symmetry for graphs of functions of the form y = ax 2 + c. The value of c translates the graph up or down.
17 Graphing y = ax 2 + c What is the relationship of the following graphs? 1. y = 2x and y = 2x 2 2. y = x 2 and y = x y = 1 2 x2 and y = 1 2 x2 + 1
18 Practice Graph each function. 1. f x = x f x = x f x = 1 2 x2 + 2
19 Vocabulary As an object falls, its speed continues to increase, so its height above the ground decreases at a faster and faster rate. Ignoring air resistance, you can model the object s height with the function h = 16t 2 + c. The height h is in feet, the time t is in seconds, and the object s initial height c is in feet.
20 Using the Falling Object Model An acorn drops from a tree branch 20 feet above the ground. The function h = 16t gives the height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what time does the the acorn hit the ground? t h = 16t
21 Using the Falling Object Model An acorn drops from a tree branch 70 feet above the ground. The function h = 16t gives the height h of the acorn (in feet) after t seconds. What is the graph of this quadratic function? At what time does the the acorn hit the ground? t h = 16t
22 Practice A person walking across a bridge accidentally drops an orange into the river below from a height of 40 feet. The function h = 16t gives the orange s approximate height h above the water, in feet, after t seconds. In how many seconds will the orange hit the water? A bird drops a stick to the ground from a height of 80 feet. The function h = 16t gives the stick s approximate height h above the ground, in feet, after t seconds. Graph the function. At about what time does the stick hit the ground?
23 Quadratic Functions Objective: To graph quadratic functions of the form y = ax 2 + bx + c.
24 Objectives I can graph y = ax 2 + bx + c. I can use the vertical motion model.
25 Vocabulary In the quadratic function y = ax 2 + bx + c, the value of b affects the position of the axis of symmetry. The axis of symmetry changes with each equation because of the change in the bvalue. The equation of the axis of symmetry is related to the ratio b a. The equation of the axis of symmetry is x = 1 2 b a or x = b 2a. Graph of a Quadratic Function o The graph of y = ax 2 + bx + c, where a 0, has the line x = b The x coordinate of the vertex is b 2a. 2a as its axis of symmetry.
26 Vocabulary When you substitute x = 0 into the equation y = ax 2 + bx + c, you get y = c. So the y intercept of a quadratic function is c. You can use the axis of symmetry and the y intercept to help you graph a quadratic function.
27 Graphing y = ax 2 + bx + c What is the graph of the function? Show the axis of symmetry. 1. y = x 2 6x y = x 2 + 4x 2 3. y = 2x y = 3x x f x = x 2 + 4x 5 6. f x = 4x
28 Practice What is the graph of the function? Show the line of symmetry. 1. y = 2x 2 6x f x = 2x 2 + 4x 1 3. y = 6x 2 + 6x 5 4. f x = 5x 2 + 3x y = 2x 2 10x 6. y = 4x 2 16x 3
29 Vocabulary You have used h = 16t 2 + c to find the height h above the ground of an object falling from an initial height c at time t. If an object projected into the air given an initial upward velocity v continues with no additional force of its own, the formula h = 16t 2 + vt + c givens its approximate height above the ground.
30 Using a Vertical Motion Model During halftime of a basketball game, a sling shot launches T shirts at the crowd. A T shirt launched with an initial upward velocity of 72 feet per second. The T shirt is caught 35 feet above the court. The T shirt is launched from a height of 5 feet. a. How long will it take the T shirt to reach its maximum height? b. What is the maximum height? c. What is the range of the function that models the height of the T shirt over time?
31 Using a Vertical Motion Model During halftime of a basketball game, a sling shot launches T shirts at the crowd. A T shirt launched with an initial upward velocity of 64 feet per second. The T shirt is caught 35 feet above the court. The T shirt is launched from a height of 5 feet. a. How long will it take the T shirt to reach its maximum height? b. What is the maximum height? c. What is the range of the function that models the height of the T shirt over time?
32 Practice A baseball is thrown into the air with an upward velocity of 30 feet per second. Its height h, in feet, after t seconds is given by the function h = 16t t + 6. a. How long will it take the ball to reach its maximum height? b. What is the ball s maximum height? c. What is the range of the function?
33 Solving Quadratic Equations Objective: To solve quadratic equations by graphing and using square roots.
34 Objectives I can solve by graphing. I can solve using square roots. I can choose a reasonable solution.
35 Vocabulary Standard Form of a Quadratic Equation: A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a 0. This form is called the standard form of a quadratic equation. Quadratic equations can be solved by a variety of methods, including graphing and finding square roots. One way to solve a quadratic equation ax 2 + bx + c = 0 is to graph the related quadratic function y = ax 2 +bx + c. The solutions of the equation are the x intercepts of the related function.
36 Vocabulary A quadratic equation can have two, one, or no realnumber solutions. The solutions of a quadratic equation and the x intercepts of the graph of the related function are often called roots of the equation or zeros of the function.
37 Solving by Graphing What are the solutions of each equation? Use a graph of the related function. 1. x 2 1 = 0 2. x 2 = 0 3. x = 0 4. x 2 16 = x = 0 6. x 2 25 = 25
38 Practice Solve each equation by graphing the related function. If the equation has no real number solution, write no solution. 1. x 2 9 = x 2 = x2 3 = 0 4. x = 5 5. x 2 10 = x 2 18 = 0
39 Vocabulary You can solve equations of the form x 2 = k by finding the square roots of each side.
40 Solving Using Square Roots What are the solutions? 1. 3x 2 75 = 0 2. m 2 36 = x = d = t 2 = y = 0 7. x 2 25 = x 2 8 = 0
41 Practice Solve each equation by finding square roots. If the question has no real number solution, write no solution. 1. n 2 = w 2 36 = b 2 = q 2 20 = p 2 = a = 0 7. r = k = 0
42 Vocabulary You can solve some quadratic equations that model real world problems by finding square roots. In many cases, the negative square root may not be a reasonable solution.
43 Choosing a Reasonable Solution 1. An aquarium is designing a new exhibit to showcase tropical fish. The exhibit will include a tank that is a rectangular prism with a length l that is twice the width w. The volume of the tank is 420 ft 3. What is the width of the tank to the nearest tenth of a foot? (V = lwh), h = 3 ft 2. Suppose that the tank has a height of 4 feet and a volume of 500 ft 3. What is the width of the tank to the nearest tenth of a foot.
44 Practice You have enough paint to cover an area of 50 ft². What is the side length of the largest square that you could paint? Round your answer to the nearest tenth of a foot. Find the length of a square with an area of 75 ft². Round to your answer to the nearest tenth of a foot.
45 Factoring to Solve Quadratic Equations Objective: To solve quadratic equations by factoring.
46 Objectives I can use the zeroproduct property. I can solve by factoring. I can write in standard form first. I can use factoring to solve realworld problems.
47 Vocabulary In the previous lesson, you solved quadratic equations ax 2 + bx + c = 0 by finding square roots. This method works if b = 0. You can solve some quadratic equations, including equations where b 0, by using the Zero Product Property. The Multiplication Property of Zero states that for any real number a, a 0 = 0. This is equivalent to the following statement: For any real numbers a and b, if a = 0 or b = 0, then ab = 0. The Zero Product Property reverses this statement.
48 Vocabulary Zero Product Property For any real numbers a and b, if ab = 0, then a = 0 or b = 0. Example: If (x + 3)(x + 2) = 0, then x + 3 = 0 or x + 2 = 0.
49 Using the Zero Product Property What are the solutions of the equation? 1. (4t + 1)(t 2) = 0 2. (x + 1)(x 5) = 0 3. (2x + 3)(x 4) = 0 4. (2y + 1)(y + 14) = 0 5. (7n 2)(5n 4) = 0 6. (v 4)(v 7) = 0
50 Practice Use the Zero Product Property to solve each equation. 1. (x 9)(x 8) = 0 2. (4k + 5)(k + 7) = 0 3. n(n + 2) = n(2n 5) = 0 5. (7x + 2)(5x 4) = 0 6. (4a 7)(3a + 8) = 0
51 Vocabulary You can also use the Zero Product Property to solve equations of the form ax 2 + bx + c = 0 if the quadratic expression ax 2 + bx + c can be factored.
52 Solving by Factoring What is the solutions of the equations? 1. x 2 + 8x + 15 = 0 2. m 2 5m 14 = 0 3. p 2 + p 20 = a 2 15a + 18 = 0 5. t 2 + 3t 54 = y 2 17y + 24 = 0
53 Practice Solve by factoring. 1. x x + 10 = 0 2. g 2 + 4g 32 = q 2 + q 14 = 0 4. p 2 4p = w 2 11w = b 2 = 81
54 Vocabulary Before solving a quadratic equation, you may need to add or subtract terms from each side in order to write the equation in standard form. Then factor the quadratic expression.
55 Writing in Standard Form First What are the solutions? 1. 4x 2 21x = x x = p 2 4p = w 2 11w = h h = b 2 = 16
56 Practice Solve by factoring. 1. x x = c 2 = 5c 3. t 2 = 3t y 2 17y = n n + 15 = 2n q 2 + 3q = 3q 2 4q + 18
57 Using Factoring to Solve a Real World Problem You are constructing a frame for a rectangular photo (17 in by 11 in).you want the frame to be the same width all the way around and the total area of the frame and photo to be 315 in 2. What should the outer dimensions of the frame be?
58 Using Factoring to Solve a Real World Problem You are constructing a frame for a rectangular photo (17 in by 11 in).you want the frame to be the same width all the way around and the total area of the frame and photo to be 391 in 2. What should the outer dimensions of the frame be?
59 Practice A box shaped like a rectangular prism has a volume of 280 in³. Its dimensions are 4 in by (n + 2) in by (n + 5) in. Find n. You are building a rectangular deck. The area of the deck should be 250 ft 2. You want the length of the deck to be 5 feet longer than twice its width. What should the dimensions of the deck be?
60 Completing the Square Objective: To solve quadratic equations by completing the square.
61 Objective I can find c to complete the square. I can solve x 2 + bx + c = 0. I can find the vertex by completing the square. I can complete the square when a 1.
62 Vocabulary You can solve any quadratic equation by first writing it in the form m 2 = n. In general, you can change the expression x 2 + bx into a perfect square trinomial by adding ( b 2 )2 to x 2 + bx. This process is called completing the square. The process is the same whether b is positive or negative.
63 Finding c to Complete the Square What is the value of c so that it is a perfect square trinomial? 1. x 2 16x + c 2. x x + c 3. g g + c 4. q 2 4q + c 5. k 2 5k + c 6. x x + c
64 Practice Find the value of c such that each expression is a perfect square trinomial. 1. x 2 + 4x + c 2. a 2 7a + c 3. b b + c 4. w w + c 5. g 2 20g + c 6. n 2 9n + c
65 Vocabulary To solve an equation in the form x2 + bx + c = 0, first subtract the constant term c from each side of the equation.
66 Solving x 2 + bx + c = 0 What is the solution of the equation? 1. x 2 14x + 16 = 0 2. x 2 + 9x + 15 = 0 3. m 2 + 7m 294 = 0 4. m m = g 2 + 7g = z 2 2z = 323
67 Practice What are the solutions for the following? 1. x 2 + 6x = t 2 6t = r 2 4r = p 2 + 5p 7 = 0 5. m m + 19 = 0 6. w 2 14w + 13 = 0
68 Vocabulary The equation y = (x h) 2 +k represents a parabola with vertex (h, k). You can use the method of completing the square to find the vertex of quadratic functions of the form y = x 2 + bx + c.
69 Finding the Vertex by Completing the Square Find the vertex by completing the square. 1. y = x 2 + 6x y = x 2 + 4x y = x x y = x x y = x x 468
70 Practice Find the vertex of each parabola by completing the square. 1. y = x 2 + 4x y = x 2 + 6x 7 3. y = x 2 + 2x y = x 2 2x 323
71 Vocabulary The method of completing the square works when a = 1 in ax 2 + bx + c = 0. To solve an equation when a 1, divide each side by a before completing the square.
72 Completing the Square When a 1 You are planning a flower garden consisting of three square plots surrounded by a 1 foot border. The total area of the garden and the border is 100 ft 2. What is the side length x of each square? 1 x 1 1 x x 1 x
73 Completing the Square When a 1 You are planning a flower garden consisting of three square plots surrounded by a 1 foot border. The total area of the garden and the border is 150 ft 2. What is the side length x of each square? Round to the nearest hundredth. 1 1 x x x x
74 Practice Solve each equation by completing the square. If necessary, round to the nearest hundredth. 1. 4a 2 8a = y 2 8y 10 = n 2 3n 15 = w w 44 = r r = v 2 10v 20 = 8
75 The Quadratic Formula and the Discriminant Objective: To solve quadratic equations using the quadratic formula. To find the number of solutions of a quadratic equation.
76 Objectives I can use the quadratic formula. I can find approximate solutions. I can choose an appropriate method. I can use the discriminant.
77 Vocabulary You can find the solution(s) of any quadratic equation using the quadratic formula. Quadratic Formula: Algebra: If ax2 + bx + c = 0, and a 0, then x = b± b2 4ac. 2a Example: Suppose 2x 2 + 3x 5 = 0. Then a = 2, b = 3, and c = 5. Therefore x = (3)± (3)2 4(2)( 5) 2(2) Be sure to write a quadratic in standard form before using the quadratic formula.
78 Here s Why It Works: If you complete the square for the general equation ax 2 + bx + c = 0, you can derive the quadratic formula. Step 1: Write ax 2 + bx + c = 0 so that the coefficient of x 2 is 1. ax 2 + bx + c = 0 x 2 + b x + c = 0 Divide each side by a. a a Step 2: Complete the square. x 2 + b x = c a a x 2 + b b x + ( a 2a )2 = c + ( b a 2a )2 (x + b 2a )2 = c + b2 a 4a 2 (x + b 2a )2 = 4ac 4a 2 + b2 4a 2 (x + b 2a )2 = b2 4ac 4a 2 Step 3: Solve the equation for x. (x + b 2a )2 =± x + b = ± b2 4ac 2a 2a x = b ± b2 4ac 2a 2a x = b± b2 4ac 2a b2 4ac 4a 2 Subtract c from each side. a Add ( b 2a )2 to each side. Write the left side as a square. Multiply c a Simplify the right side. 4a by to get like denominators. 4a Take square roots of each side. Simplify the right side. Subtract b from each side. 2a Simplify
79 Using the Quadratic Formula What are the solutions using the quadratic formula? 1. x 2 8 = 2x 2. x 2 4x = x 2 + 5x + 3 = x x = x 2 45x 50 = 0
80 Practice Use the quadratic formula to solve each equation. 1. 3x 2 41x = x x 84 = x 2 x 120 = x x = x 2 47x = 156
81 Vocabulary When the radicand in the quadratic formula is not a perfect square, you can use a calculator to approximate the solutions of an equation.
82 Finding Approximate Solutions 1. In the shot put, an athlete throws a heavy metal ball through the air. The arc of the ball can be modeled by the equation y = 0.04x x + 2, where x is the horizontal distance, in meters, from the athlete and y is the height, in meters, of the ball. How far from the athlete will the ball land? 2. A batter strikes a baseball. The equation y = 0.005x x models its path, where x is the horizontal distance, in feet, the ball travels and y is the height, in feet, of the ball. How far from the batter will the ball land? Round to the nearest tenth of a foot.
83 Practice Use the quadratic formula to solve each equation. Round your answer to the nearest hundredth. 1. x 2 + 8x + 11 = x x 2 = x 2 16x = x 2 + 9x = 32
84 Vocabulary There are many methods for solving a quadratic equation. Method Graphing Square Roots Factoring Completing the Square Quadratic Formula When to Use Use if you have a graphing calculator handy Use if the equation has no xterms Use if you can factor the equation easily Use if the coefficient of x 2 is 1, but you cannot easily factor the equation Use if the equation cannot be factored easily or at all
85 Choosing an Appropriate Method Which method(s) would you choose to solve each equation? Explain your reasoning. 1. 3x 2 9 = 0 2. x 2 x 30 = x x 17 = 0 4. x 2 5x + 3 = x 2 50x + 21 = 0 6. x 2 8x + 12 = x 2 = x x 1 = 0
86 Practice Which method(s) would you choose to solve each equation? Justify your reasoning. 1. x 2 + 4x 15 = x 2 49 = x 2 41x = x 2 7x + 3 = 0 5. x 2 + 4x 60 = x 2 + 8x + 1 = 0
87 Vocabulary Quadratic equations can have two, one, or no realnumber solutions. Before you solve a quadratic equations, you can determine how many realnumber solutions it has by using the discriminant. The discriminant is the expression under the radical sign in the quadratic formula. The discriminant is b 2 4ac of x = b± b2 4ac 2a The discriminant of a quadratic equation can be positive, negative, or zero.
88 Using the Discriminant Discriminant b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 Example x 2 6x + 7 = 0 The discriminant (6) 2 4(1)(7) = 8 which is positive x 2 6x + 9 = 0 The discriminant (6) 2 4(1)(9) = 0 x 2 6x + 11 = 0 The discriminant (6) 2 4(1)(11) = 8 which is negative Number of Solutions There are two realnumber solutions There is one realnumber solutions There are no realnumber solutions
89 The Discriminant
90 Using the Discriminant How many realnumber solutions does each have? 1. 2x 2 3x = x 2 5x = 7 3. x 2 + 3x + 11 = x x + 4 = 0 5. x 2 15 = 0
91 Practice Find the number of real number solutions of each equation. 1. x 2 2x + 3 = 0 2. x 2 + 7x 5 = 0 3. x 2 + 2x = p 2 + 4p = 10
CC Algebra Quadratic Functions Test Review. 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1?
Name: CC Algebra Quadratic Functions Test Review Date: 1. The graph of the equation y = x 2 is shown below. 4. Which parabola has an axis of symmetry of x = 1? a. c. c. b. d. Which statement best describes
More information6.1 Solving Quadratic Equations by Factoring
6.1 Solving Quadratic Equations by Factoring A function of degree 2 (meaning the highest exponent on the variable is 2), is called a Quadratic Function. Quadratic functions are written as, for example,
More information98 Completing the Square
In the previous lesson, you solved quadratic equations by isolating x 2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square. When
More informationSolving Quadratic Equations: Algebraically and Graphically Read 3.1 / Examples 1 4
CC Algebra II HW #14 Name Period Row Date Solving Quadratic Equations: Algebraically and Graphically Read 3.1 / Examples 1 4 Section 3.1 In Exercises 3 12, solve the equation by graphing. (See Example
More informationChapter 1 Notes: Quadratic Functions
19 Chapter 1 Notes: Quadratic Functions (Textbook Lessons 1.1 1.2) Graphing Quadratic Function A function defined by an equation of the form, The graph is a Ushape called a. Standard Form Vertex Form
More informationChapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand
Chapter 4: Quadratic Functions and Factoring 4.1 Graphing Quadratic Functions in Stand VOCAB: a quadratic function in standard form is written y = ax 2 + bx + c, where a 0 A quadratic Function creates
More informationControlling the Population
Lesson.1 Skills Practice Name Date Controlling the Population Adding and Subtracting Polynomials Vocabulary Match each definition with its corresponding term. 1. polynomial a. a polynomial with only 1
More informationQuadratic Equations. Math 201 Chapter 4. General Outcome: Develop algebraic and graphical reasoning through the study of relations.
Math 201 Chapter 4 Quadratic Equations General Outcome: Develop algebraic and graphical reasoning through the study of relations. Specific Outcomes: RF1. Factor polynomial expressions of the form: ax
More informationLadies and Gentlemen: Please Welcome the Quadratic Formula!
Lesson.1 Skills Practice Name Date Ladies and Gentlemen: Please Welcome the Quadratic Formula! The Quadratic Formula Vocabulary Complete the Quadratic Formula. Then, identify the discriminant and explain
More informationSubtract 16 from both sides. Divide both sides by 9. b. Will the swing touch the ground? Explain how you know.
REVIEW EXAMPLES 1) Solve 9x + 16 = 0 for x. 9x + 16 = 0 9x = 16 Original equation. Subtract 16 from both sides. 16 x 9 Divide both sides by 9. 16 x Take the square root of both sides. 9 4 x i 3 Evaluate.
More informationMaintaining Mathematical Proficiency
Chapter Maintaining Mathematical Proficiency Simplify the expression. 1. 8x 9x 2. 25r 5 7r r + 3. 3 ( 3x 5) + + x. 3y ( 2y 5) + 11 5. 3( h 7) 7( 10 h) 2 2 +. 5 8x + 5x + 8x Find the volume or surface area
More informationy ax bx c OR 0 then either a = 0 OR b = 0 Steps: 1) if already factored, set each factor in ( ) = 0 and solve
Algebra 1 SOL Review: Quadratics Name 67B Solving Quadratic equations using ZeroProduct Property. Quadratic equation: ax bx c 0 OR y ax bx c OR f ( x ) ax bx c ZeroProduct Property: if a b 0 then either
More informationName Teacher: Period: Date: ALGEBRA I FINAL REVIEW SPRING 2016
Name Teacher: Period: Date: ALGEBRA I FINAL REVIEW SPRING 016 Solve each system of inequalities by graphing. (Book sections 45and 46) y x4 1.).) y x1 y x 4 4yx.) x y 14 x y0 4.) x y 4x y 10 5.) x y 1
More informationUnit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #2: THE PARABOLA APPLICATIONS AND WORD PROBLEMS INVERSE OF A QUADRATIC FUNCTION DO NOW: Review from Lesson #1 (a)using the graph shown to the right, determine the equation
More informationLesson 10.1 Solving Quadratic Equations
Lesson 10.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with each set of conditions. a. One intercept and all nonnegative yvalues b. The verte in the third quadrant and no
More informationMidChapter Quiz: Lessons 11 through 14
Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every xvalue there is exactly one corresponding yvalue. function
More informationQuadratic Equations Chapter Questions
Quadratic Equations Chapter Questions 1. Describe the characteristics of a quadratic equation. 2. What are the steps for graphing a quadratic function? 3. How can you determine the number of solutions
More informationSummer Prep Packet for students entering Algebra 2
Summer Prep Packet for students entering Algebra The following skills and concepts included in this packet are vital for your success in Algebra. The Mt. Hebron Math Department encourages all students
More informationMission 1 Factoring by Greatest Common Factor and Grouping
Algebra Honors Unit 3 Factoring Quadratics Name Quest Mission 1 Factoring by Greatest Common Factor and Grouping Review Questions 1. Simplify: i(6 4i) 3+3i A. 4i C. 60 + 3 i B. 8 3 + 4i D. 10 3 + 3 i.
More information1) Explain in complete sentences how to solve the following equation using the factoring method. Y=7x
TEST 13 REVIEW Quadratics 1) Explain in complete sentences how to solve the following equation using the factoring method. Y=7x 2 +28. 2) Find the domain and range if the points in the table are discrete
More informationFinal Review. Intermediate Algebra / MAT135 S2014
Final Review Intermediate Algebra / MAT135 S2014 1. Solve for. 2. Solve for. 3. Solve for. 4. Jenny, Abdul, and Frank sent a total of text messages during the weekend. Abdul sent more messages than Jenny.
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More information. State the important connection between the coefficients of the given trinomials and the values you found for r.
Motivational Problems on Quadratics 1 1. Factor the following perfectsquare trinomials : (a) x 1x 36 (b) x 14x 49 (c) x 0x 100 As suggested, these should all look like either ( x r) or ( x r). State the
More informationTuesday, 3/28 : Ch. 9.8 Cubic Functions ~ Ch. 9 Packet p.67 #(16) Thursday, 3/30 : Ch. 9.8 Rational Expressions ~ Ch. 9 Packet p.
Ch. 9.8 Cubic Functions & Ch. 9.8 Rational Expressions Learning Intentions: Explore general patterns & characteristics of cubic functions. Learn formulas that model the areas of squares & the volumes of
More information1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = 2x 2 4x + 2 x y.
1. Graph (on graph paper) the following equations by creating a table and plotting points on a coordinate grid y = 2x 2 4x + 2 x y y = x 2 + 6x 3 x y domain= range= 43 21 0 1 2 3 4 domain= range=
More informationImportant Math 125 Definitions/Formulas/Properties
Exponent Rules (Chapter 3) Important Math 125 Definitions/Formulas/Properties Let m & n be integers and a & b real numbers. Product Property Quotient Property Power to a Power Product to a Power Quotient
More informationAlgebra Notes Quadratic Functions and Equations Unit 08
Note: This Unit contains concepts that are separated for teacher use, but which must be integrated by the completion of the unit so students can make sense of choosing appropriate methods for solving quadratic
More information1) Solve the quadratic equation Y=5x*+3 where *=2 A. x = (Y3) B. x = (3+Y) C. x = (3+Y) 2 D. x = (Y3) 2
TEST 13 REVIEW Quadratics 1) Solve the quadratic equation Y=5x*+3 where *=2 A. x = (Y3) B. x = (3+Y) C. x = (3+Y) 2 D. x = (Y3) 2 2) Explain in complete sentences how to solve the following equation
More informationQuadratic Functions. and Equations
Name: Quadratic Functions and Equations 1. + x 2 is a parabola 2.  x 2 is a parabola 3. A quadratic function is in the form ax 2 + bx + c, where a and is the yintercept 4. Equation of the Axis of Symmetry
More informationLesson 9 Exploring Graphs of Quadratic Functions
Exploring Graphs of Quadratic Functions Graph the following system of linear inequalities: { y > 1 2 x 5 3x + 2y 14 a What are three points that are solutions to the system of inequalities? b Is the point
More informationAlgebra II 5.3 Solving Quadratic Equations by Finding Square Roots
5.3 Solving Quadratic Equations by Finding Square Roots Today I am solving quadratic equations by finding square roots. I am successful today when solve quadratic functions using square roots. It is important
More informationProperties of Radicals
9. Properties of Radicals Essential Question How can you multiply and divide square roots? Operations with Square Roots Work with a partner. For each operation with square roots, compare the results obtained
More information3.1. Have you ever seen a tightrope walker? If you ve ever seen this, you know that it. Shape and Structure. Forms of Quadratic Functions
Shape and Structure Forms of Quadratic Functions.1 Learning Goals In this lesson, you will: Match a quadratic function with its corresponding graph. Identify key characteristics of quadratic functions
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Rationalize the denominator and simplify. 1 1) B) C) 1 D) 1 ) Identify the pair of like
More informationr r 30 y 20y 8 7y x 6x x 5x x 8x m m t 9t 12 n 4n r 17r x 9x m 7m x 7x t t 18 x 2x U3L1  Review of Distributive Law and Factoring
UL  Review of Distributive Law and Factoring. Expand and simplify. a) (6mn )(5m 4 n 6 ) b) 6x 4 y 5 z 7 (x 7 y 4 z) c) (x 4)  (x 5) d) (y 9y + 5) 5(y 4) e) 5(x 4y) (x 5y) + 7 f) 4(a b c) 6(4a + b
More informationUnit 21: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions
CP Algebra Unit 1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.
More information3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
More informationUNIT PLAN: EXPLORING QUADRATIC FUNCTIONS. graph quadratic equations with the use of technology.
Ashley Swift February 15, 2001 UNIT PLAN: EXPLORING QUADRATIC FUNCTIONS UNIT OBJECTIVES: This unit will help students learn to: graph quadratic equations with the use of technology. find the axis of symmetry
More informationChapter 2 Polynomial and Rational Functions
SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear
More informationMATH College Algebra Review for Test 2
MATH 4  College Algebra Review for Test Sections. and.. For f (x) = x + 4x + 5, give (a) the xintercept(s), (b) the intercept, (c) both coordinates of the vertex, and (d) the equation of the axis of
More information1Add and subtract 2Multiply radical
Then You simplified radical expressions. (Lesson 102) Now 1Add and subtract radical expressions. 2Multiply radical expressions. Operations with Radical Expressions Why? Conchita is going to run in her
More informationSimplifying Radicals. multiplication and division properties of square roots. Property Multiplication Property of Square Roots
102 Simplifying Radicals Content Standard Prepares for A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Objective To simplify
More informationMAT 1033C  MartinGay Intermediate Algebra Chapter 8 (8.1, 8.2, 8.5, 8.6) Practice for the Exam
MAT 33C  MartinGa Intermediate Algebra Chapter 8 (8.1 8. 8. 8.6) Practice for the Eam Name Date Da/Time: SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
More informationSolving Quadratic Equations by Formula
Algebra Unit: 05 Lesson: 0 Complex Numbers All the quadratic equations solved to this point have had two real solutions or roots. In some cases, solutions involved a double root, but there were always
More informationLesson 8 Solving Quadratic Equations
Lesson 8 Solving Quadratic Equations Lesson 8 Solving Quadratic Equations We will continue our work with quadratic equations in this lesson and will learn the classic method to solve them the Quadratic
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PHYS 101 Fall 2013 (Purcell), Fake Midterm #1 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) The figure shows the graph of the position x as a
More informationChapter 3: Polynomial and Rational Functions
3.1 Power and Polynomial Functions 155 Chapter 3: Polynomial and Rational Functions Section 3.1 Power Functions & Polynomial Functions... 155 Section 3. Quadratic Functions... 163 Section 3.3 Graphs of
More informationTips for doing well on the final exam
Algebra I Final Exam 01 Study Guide Name Date Block The final exam for Algebra 1 will take place on May 1 and June 1. The following study guide will help you prepare for the exam. Tips for doing well on
More informationSolve each equation by completing the square. Round to the nearest tenth if necessary. 5. x 2 + 4x = 6 ANSWER: 5.2, 1.2
Find the value of c that makes each trinomial a perfect square. 1. x 2 18x + c 81 3. x 2 + 9x + c Solve each equation by completing the square. Round to the nearest tenth if necessary. 5. x 2 + 4x = 6
More informationPHY 1114: Physics I. Quick Question 1. Quick Question 2. Quick Question 3. Quick Question 4. Lecture 5: Motion in 2D
PHY 1114: Physics I Lecture 5: Motion in D Fall 01 Kenny L. Tapp Quick Question 1 A child throws a ball vertically upward at the school playground. Which one of the following quantities is (are) equal
More informationFinal Exam Review Part 2 #4
Final Eam Review Part # Intermediate Algebra / MAT 135 Fall 01 Master (Prof. Fleischner) Student Name/ID: 1. Solve for, where is a real number. + = 8. Solve for, where is a real number. 9 1 = 3. Solve
More informationAn equation is a statement that states that two expressions are equal. For example:
Section 0.1: Linear Equations Solving linear equation in one variable: An equation is a statement that states that two expressions are equal. For example: (1) 513 (2) 16 (3) 4252 (4) 64153 To solve the
More informationLesson 17 Quadratic Word Problems. The equation to model Vertical Motion is
W8D1 Quadratic Word Problems Warm Up 1. A rectangle has dimensions of x+2 and x+3. What is the area of the rectangle? 2. What is the Perimeter of the rectangle? 3. If the area of the rectangle is 30 m
More informationMAT Intermediate Algebra  Final Exam Review Textbook: Beginning & Intermediate Algebra, 5th Ed., by MartinGay
MAT0  Intermediate Algebra  Final Eam Review Tetbook: Beginning & Intermediate Algebra, 5th Ed., by MartinGay Section 2. Solve the equation. ) 9  (  ) = 2 Section 2.8 Solve the inequality. Graph the
More informationAlgebra II Summer Review To earn credit: Be sure to show all work in the area provided.
Summer Review for Students Who Will Take Algebra II in 20162017 1 Algebra II Summer Review To earn credit: Be sure to show all work in the area provided. Evaluating and Simplifying Applying Order of Operations:
More information1 Which expression represents 5 less than twice x? 1) 2) 3) 4)
1 Which expression represents 5 less than twice x? 2 Gabriella has 20 quarters, 15 dimes, 7 nickels, and 8 pennies in a jar. After taking 6 quarters out of the jar, what will be the probability of Gabriella
More informationCh2 practice test. for the following functions. f (x) = 6x 2 + 2, Find the domain of the function using interval notation:
Ch2 practice test Find for the following functions. f (x) = 6x 2 + 2, Find the domain of the function using interval notation: A hotel chain charges $75 each night for the first two nights and $55 for
More informationStudy Resources For Algebra I. Unit 2A Graphs of Quadratic Functions
Study Resources For Algebra I Unit 2A Graphs of Quadratic Functions This unit examines the graphical behavior of quadratic functions. Information compiled and written by Ellen Mangels, Cockeysville Middle
More informationPRINCIPLES OF MATHEMATICS 11 Chapter 2 Quadratic Functions Lesson 1 Graphs of Quadratic Functions (2.1) where a, b, and c are constants and a 0
PRINCIPLES OF MATHEMATICS 11 Chapter Quadratic Functions Lesson 1 Graphs of Quadratic Functions (.1) Date A. QUADRATIC FUNCTIONS A quadratic function is an equation that can be written in the following
More informationAlgebra 1. Correlated to the Texas Essential Knowledge and Skills. TEKS Units Lessons
Algebra 1 Correlated to the Texas Essential Knowledge and Skills TEKS Units Lessons A1.1 Mathematical Process Standards The student uses mathematical processes to acquire and demonstrate mathematical understanding.
More informationMATH 8 CATALINA FOOTHILLS SCHOOL DISTRICT
MATH 8 CATALINA FOOTHILLS SCHOOL DISTRICT Overarching Understandings for the Course: Students will understand Number Systems, Percents, Expressions, Equations, and Inequalities, Exponent Rules, Functions,
More informationName Date Class , 100, 1000, 10,000, common ratio:
Name Date Class 111 Practice A Geometric Sequences Find the common ratio of each geometric sequence. Then find the next three terms in each geometric sequence. 1. 1, 4, 16, 64, 2. 10, 100, 1000, 10,000,
More information5) A stone is thrown straight up. What is its acceleration on the way up? 6) A stone is thrown straight up. What is its acceleration on the way down?
5) A stone is thrown straight up. What is its acceleration on the way up? Answer: 9.8 m/s 2 downward 6) A stone is thrown straight up. What is its acceleration on the way down? Answer: 9.8 m/ s 2 downward
More informationHonors Algebra 2 ~ Spring 2014 Name 1 Unit 3: Quadratic Functions and Equations
Honors Algebra ~ Spring Name Unit : Quadratic Functions and Equations NC Objectives Covered:. Define and compute with comple numbers. Operate with algebraic epressions (polnomial, rational, comple fractions)
More informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationSolve each equation by using the Square Root Property. Round to the nearest hundredth if necessary.
1. Solve each equation by using the Square Root Property. Round to the nearest hundredth if necessary. 2. 3. 4. 5. LASER LIGHT SHOW The area A in square feet of a projected laser light show is given by
More informationName Class Date. Quadratic Functions and Transformations. 4 6 x
 Quadratic Functions and Transformations For Eercises, choose the correct letter.. What is the verte of the function 53()? D (, ) (, ) (, ) (, ). Which is the graph of the function f ()5(3) 5? F 6 6 O
More informationALGEBRA I ENDOFCOURSE EXAM: PRACTICE TEST
Page 1 ALGEBRA I ENDOFCOURSE EXAM: PRACTICE TEST 1. Order the following numbers from least to greatest:, 6, 8.7 10 0, 19 b. 19,, 8.7 100, 6 6, 8.7 10 0,, 19 c. d. 8.7 10 0,, 19, 6, 6, 19, 8.7 100. If
More informationAlgebra I Assessment. Eligible Texas Essential Knowledge and Skills
Algebra I Assessment Eligible Texas Essential Knowledge and Skills STAAR Algebra I Assessment Mathematical Process Standards These student expectations will not be listed under a separate reporting category.
More informationNew Jersey Quality Single Accountability Continuum (NJQSAC) ASSE 12; ACED 1,4; AREI 13, FIF 15, 7a
ALGEBRA 2 HONORS Date: Unit 1, September 430 How do we use functions to solve real world problems? What is the meaning of the domain and range of a function? What is the difference between dependent variable
More informationx and y, called the coordinates of the point.
P.1 The Cartesian Plane The Cartesian Plane The Cartesian Plane (also called the rectangular coordinate system) is the plane that allows you to represent ordered pairs of real numbers by points. It is
More informationDate: 20 (month/day/year) Amount of Time to Complete (to be filled in by your teacher): mins. Instructions
James J. Kaput Center for Research and Innovation in Mathematics Education 200 Mill Rd., Suite 150B Fairhaven, MA 02719, USA Full Name: Date: 20 (month/day/year) Teacher Name: Amount of Time to Complete
More information5. 2. The solution set is 7 6 i, 7 x. Since b = 20, add
Chapter : Quadratic Equations and Functions Chapter Review Eercises... 5 8 6 8 The solution set is 8, 8. 5 5 5 5 5 5 The solution set is 5,5. Rationalize the denominator. 6 The solution set is. 8 8 9 6
More informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More informationPreCalc Chapter 1 Sample Test. D) slope: 3 4
PreCalc Chapter 1 Sample Test 1. Use the graphs of f and g to evaluate the function. f( x) gx ( ) (f o g)(0.5) 1 1 0 4. Plot the points and find the slope of the line passing through the pair of points.
More informationBasic Equation Solving Strategies
Basic Equation Solving Strategies Case 1: The variable appears only once in the equation. (Use work backwards method.) 1 1. Simplify both sides of the equation if possible.. Apply the order of operations
More informationINTRODUCTION. 3. TwoDimensional Kinematics
INTRODUCTION We now extend our study of kinematics to motion in two dimensions (x and y axes) This will help in the study of such phenomena as projectile motion Projectile motion is the study of objects
More informationAdding Vectors in Two Dimensions
Slide 37 / 125 Adding Vectors in Two Dimensions Return to Table of Contents Last year, we learned how to add vectors along a single axis. The example we used was for adding two displacements. Slide 38
More informationClass: Date: ID: A. 1, Write the polynomial so that the exponents decrease from left to right. c. llb~4b 26b+3 d.
Class: Date: ID: A A~gebra 1  Fina~ Exam Review 2(H3 1, Write the polynomial so that the exponents decrease from left to right. 6X 3  6X + 4x s  2 4x5 +6x3 _6x_2 c. 4x 56x3 +6x+2 b. 2+4x 56x+6x3
More informationACTIVITY: Factoring Special Products. Work with a partner. Six different algebra tiles are shown below.
7.9 Factoring Special Products special products? How can you recognize and factor 1 ACTIVITY: Factoring Special Products Work with a partner. Six different algebra tiles are shown below. 1 1 x x x 2 x
More informationSection 3.1 Exercises
.1 Power and Polynomial Functions 44 Section.1 Exercises Find the long run behavior of each function as x and x 4 6 1. f x x. f x x. f x x 5 4. f x x 4 7 9 5. f x x 6. f x x 7. f x x 8. f x x Find the
More informationAlgebra II: Chapter 4 Semester Review Multiple Choice: Select the letter that best answers the question. D. Vertex: ( 1, 3.5) Max. Value: 1.
Algebra II: Chapter Semester Review Name Multiple Choice: Select the letter that best answers the question. 1. Determine the vertex and axis of symmetry of the. Determine the vertex and the maximum or
More informationProjectile Motion. v = v 2 + ( v 1 )
What do the following situations have in common? Projectile Motion A monkey jumps from the branch of one tree to the branch of an adjacent tree. A snowboarder glides at top speed off the end of a ramp
More informationSTANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II. 2 nd Nine Weeks,
STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 2 nd Nine Weeks, 20162017 1 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource
More informationUnit 1, Pretest Functions. Advanced Mathematics PreCalculus
Unit 1, Pretest Functions Advanced Mathematics PreCalculus Blackline Masters, Advanced Math PreCalculus Page 1 Louisiana Comprehensive Curriculum, Revised 008 Most of the math symbols in this document
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y  4)  (y + ) = 3y 1) A)
More informationSolving Systems of Linear and Quadratic Equations
9.5 Solving Systems of Linear and Quadratic Equations How can you solve a system of two equations when one is linear and the other is quadratic? ACTIVITY: Solving a System of Equations Work with a partner.
More informationSection 5: Quadratic Equations and Functions Part 1
Section 5: Quadratic Equations and Functions Part 1 Topic 1: RealWorld Examples of Quadratic Functions... 121 Topic 2: Factoring Quadratic Expressions... 125 Topic 3: Solving Quadratic Equations by Factoring...
More informationWhich of the following points could be removed so that Andrew s graph represents a function?
Algebra 1 Benchmark Task Cards MA.912.A.2.3 BENCHMARK: MA.912.A.2.3 Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions.
More information5 Projectile Motion. Projectile motion can be described by the horizontal and vertical components of motion.
Projectile motion can be described by the horizontal and vertical components of motion. In the previous chapter we studied simple straightline motion linear motion. Now we extend these ideas to nonlinear
More informationnonadjacent angles that lie on opposite sides of the transversal and between the other two lines.
WORD: Absolute Value DEFINITION: The distance from zero, distance is always positive. WORD: Absolute Value Function DEFINITION: A function whose rule contains an absolute value expression. EXAMPLE(S) COUNTEREXAMPLE(S):
More informationGrade 8 Math Curriculum Map Erin Murphy
Topic 1 Variables and Expressions 2 Weeks Summative Topic Test: Students will be able to (SWBAT) use symbols o represent quantities that are unknown or that vary; demonstrate mathematical phrases and realworld
More information12.2 Simplifying Radical Expressions
Name Class Date 1. Simplifying Radical Expressions Essential Question: How can you simplify expressions containing rational exponents or radicals involving nth roots? Explore A.7.G Rewrite radical expressions
More informationAlgebra 1: Final Exam Review 2013 Mrs. Brennan!Mr. Carell
Algebra 1: Final Exam Review 2013 Mrs. Brennan!Mr. Carell Monday June 24, 2013 8:00am to 10:00am Format: 80 multiple choice 4 free response Total 1 point each 5 points each 1 O0 points Content: Chapter
More informationLesson 7.1 Polynomial Degree and Finite Differences
Lesson 7.1 Polynomial Degree and Finite Differences 1. Identify the degree of each polynomial. a. 1 b. 0.2 1. 2 3.2 3 c. 20 16 2 20 2. Determine which of the epressions are polynomials. For each polynomial,
More informationCollege Algebra with Trigonometry
College Algebra with Trigonometry This course covers the topics outlined below. You can customize the scope and sequence of this course to meet your curricular needs. Curriculum (556 topics + 614 additional
More informationFunction Junction: Homework Examples from ACE
Function Junction: Homework Examples from ACE Investigation 1: The Families of Functions, ACE #5, #10 Investigation 2: Arithmetic and Geometric Sequences, ACE #4, #17 Investigation 3: Transforming Graphs,
More informationthat when friction is present, a is needed to keep an object moving. 21. State Newton s first law of motion.
Chapter 3 Newton s First Law of Motion Inertia Exercises 31 Aristotle on Motion (pages 29 30) Fill in the blanks with the correct terms 1 Aristotle divided motion into two types: and 2 Natural motion on
More information