Section 7.1 Video Guide Solving Quadratic Equations by Completing the Square Objectives: 1. Solve Quadratic Equations Using the Square Root Property. Complete the Square in One Variable 3. Solve Quadratic Equations by Completing the Square 4. Solve Problems Using the Pythagorean Theorem Section 7.1 Objective 1: Solve Quadratic Equations Using the Square Root Property Video Length 1:1 Recall that a quadratic equation is an equation of the form,. Square Root Property If p, then or. (Note: p 0 ) 1. Eample: Solve the equation: 36 Final answer: Copyright 018 Pearson Education, Inc. 313
. Eample: Solve the equation: 3 150 Write the steps in words Step 1 Show the steps with math Step Step 3 Step 4 Final answer: 3. Eample: Solve: 3 10 Final answer: 314 Copyright 018 Pearson Education, Inc.
Section 7.1 Objective : Complete the Square in One Variable Video Length 6:09 The idea behind is to "adjust" the left side of a quadratic equation of the form b c 0 in order to make it a perfect square trinomial. Do you remember what a perfect square trinomial looks like? Obtaining a Perfect Square Trinomial Step 1: Identify the coefficient of the - term. Step : this coefficient by and then the result. Step 3: this result to both sides of the equation. Consider n 10n Let's try 16 Now try z 7z Copyright 018 Pearson Education, Inc. 315
Section 7.1 Objective 3: Solve Quadratic Equations by Completing the Square Part I Eample 6 Video Length 5:10 So let's solve a quadratic equation by completing the square. 4. Eample: Solve: 6 7 0 Write the steps in words Step 1 Show the steps with math Step Step 3 Step 4 Step 5 Note: CHECK YOUR WORK!!! Final answer: Note: You can also solve the quadratic equation get the same answer? 6 7 0 by factoring. Try it. Did you 316 Copyright 018 Pearson Education, Inc.
Section 7.1 Objective 3: Solve Quadratic Equations by Completing the Square Part II Eample 7 Video Length 7:36 5. Eample: Solve: 5 3 0 Final answer: Copyright 018 Pearson Education, Inc. 317
Section 7.1 Objective 4: Solve Problems Using the Pythagorean Theorem Video Length 4:59 The Pythagorean Theorem is a statement regarding right triangles. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. c b a 6. Eample: A baseball diamond is square. Each side of the square is 90 feet long. How far is it from home plate to second base? Draw diagram here: Final answer: 318 Copyright 018 Pearson Education, Inc.
Section 7. Video Guide Solving Quadratic Equations by the Quadratic Formula Objectives: 1. Solve Quadratic Equations Using the Quadratic Formula. Use the Discriminant to Determine the Nature of Solutions of a Quadratic Equation 3. Model and Solve Problems Involving Quadratic Equations Section 7. Objective 1: Solve Quadratic Equations Using the Quadratic Formula Part I Video Length 7:9 Proof of the Quadratic Formula: Copyright 018 Pearson Education, Inc. 319
Section 7. Objective 1: Solve Quadratic Equations Using the Quadratic Formula Part II Eample 1 Video Length 5:50 Quadratic Formula The solution(s) to the quadratic equation 0, a 0, are given by the quadratic formula a b c. Note: Each and every time you do a problem utilizing the quadratic formula, you should write it down. This way, you will ingrain the formula into your memory. 1. Eample: Solve: 11 15 Write the steps in words Step 1 Show the steps with math Step Step 3 Step 4 Final answer: 30 Copyright 018 Pearson Education, Inc.
Section 7. Objective 1: Solve Quadratic Equations Using the Quadratic Formula Part III Eample Video Length 5:38. Eample: Solve: y 4y Final answer: Copyright 018 Pearson Education, Inc. 31
Section 7. Objective 1: Solve Quadratic Equations Using the Quadratic Formula Part IV Eample 4 Video Length 8:13 3. Eample: Solve: 3m m 1 0 Final answer: 3 Copyright 018 Pearson Education, Inc.
Section 7. Objective : Use the Discriminant to Determine Which Method to Use When Solving a Quadratic Equation Video Length 10:45 Definition For a quadratic equation, the epression is called the. The discriminant is called the discriminant because it can be used to tell you not only what kind of answer you are going to get from a quadratic equation, but it also gives us a hint as to the appropriate technique to use in solving any quadratic equation. The Discriminant and Solving a Quadratic Equation For a quadratic equation, the discriminant is. 1. If, and * b 4ac is a, the quadratic equation may be solved by. * b 4ac is a, the quadratic equation is over the integers. Completing the square or the quadratic formula should be used to the solve the equation.. If, the equation may be solved by. 3. If, the equation has. 4. Eample: Solve: 7 15 0 Final answer: Copyright 018 Pearson Education, Inc. 33
5. Eample: Solve: 3 4 0 Final answer: 34 Copyright 018 Pearson Education, Inc.
Section 7. Objective 3: Model and Solve Problems Involving Quadratic Equations Video Length 7:38 6. Eample: The revenue R received by a company selling pairs of sunglasses per week is given by the function R 0.1 70. (a) Find and interpret the values of R 17 and 5 R. Final answer: (b) How many pairs of sunglasses must be sold in order for revenue to be $10,000 per week? Final answer: (c) How many pairs of sunglasses must be sold in order for revenue to be $1,50 per week? Final answer: Copyright 018 Pearson Education, Inc. 35
Section 7.3 Video Guide Solving Equations Quadratic in Form Objectives: 1. Solve Equations That are Quadratic in Form Section 7.3 Objective 1: Solve Equations That are Quadratic in Form Part I Eample 1 Video Length 9:04 Definition If a substitution u transforms an equation into one of the form then the original equation is called an. Eamples: 1. Eample: Solve: 11 18 0 4 Final answer: 36 Copyright 018 Pearson Education, Inc.
Section 7.3 Objective 1: Solve Equations That are Quadratic in Form Part II Eample 5 Video Length 8:19 Note: This is a really nice problem. It shows that it really pays to ALWAYS CHECK your answers.. Eample: Solve: 1/ 1/4 1 0 Final answer: Copyright 018 Pearson Education, Inc. 37
Objectives: Section 7.4 Video Guide Graphing Quadratic Functions Using Transformations f k 1. Graph Quadratic Functions of the Form. Graph Quadratic Functions of the Form f h 3. Graph Quadratic Functions of the Form f a 4. Graph Quadratic Functions of the Form f a b c 5. Find a Quadratic Function from Its Graph Section 7.4 Objective 1: Graph Quadratic Functions of the Form f k Video Length 13:30 Definition A is a function of the form where a, b, and c are real numbers and a 0. can be used to graph a quadratic function. 1. Eample: Graph the functions on one coordinate plane. f g f g 3 8 6 4 y 8 6 4 4 4 6 8 6 8 Notice that the graph of g is the graph of f shifted. 38 Copyright 018 Pearson Education, Inc.
. Eample: Graph the functions on one coordinate plane. f g f g 8 6 4 y 8 6 4 4 4 6 8 6 8 Notice that the graph of g is the graph of f shifted. h 3. Eample: Graph the function 4. Copyright 018 Pearson Education, Inc. 39
Section 7.4 Objective : Graph Quadratic Functions of the Form f h Video Length 11:30 4. Eample: Graph the functions on one coordinate plane. y f g 3 f g 8 6 8 4 6 4 4 4 6 8 6 8 Notice that the graph of g is the graph of f shifted. 5. Eample: Graph the functions on one coordinate plane. y f g 1 f g 8 6 8 4 6 4 4 4 6 8 6 8 330 Copyright 018 Pearson Education, Inc. Notice that the graph of g is the graph of f shifted.
6. Eample: Graph the function f 3. Copyright 018 Pearson Education, Inc. 331
Section 7.4 Objective 3: Graph Quadratic Functions of the Form f a Video Length 14:1 7. Eample: Graph the functions on one coordinate plane. y f g f g 8 6 8 4 6 4 4 4 6 8 6 8 The graph is by a factor of. 8. Eample: Graph the functions on one coordinate plane. y 1 f g f g 8 6 8 4 6 4 4 4 6 8 6 8 The graph is ( by a factor of 1 ). 33 Copyright 018 Pearson Education, Inc.
9. Eample: Graph the functions on one coordinate plane. y f g f g 8 6 g is a Notice that the graph of of the graph of 8 f. 4 6 4 4 4 6 8 6 8 Properties of the Form f a a, the graph of If 0 f a will open. In addition, if 0 a 1, the opening in the graph will be " " than that of y. If 1 a, the opening in the graph will be " " than that of y. If 0 a, the graph of f a will open. In addition, if 0 a 1, the opening in the graph will be " " than that of y. If a 1, the opening in the graph will be " " than that of y. When a 1, we say the graph is by a factor of a. When 0 a 1, we say that the graph is by a factor of a. Copyright 018 Pearson Education, Inc. 333
Graphing Using Transformations Graphing Functions of the Form f k To obtain the graph of f k from the graph of y, shift the graph of y units if k 0 and units if k 0. Graphing Functions of the Form f h To obtain the graph of f h from the graph of y, shift the graph of y to the units if h 0 and to the units if h 0. Graphing Functions of the Form f a To obtain the graph of f a from the graph of y, each on the graph of y by. 334 Copyright 018 Pearson Education, Inc.
Section 7.4 Objective 4: Graph Quadratic Functions of the Form f a b c Part I Eample 7 Video Length 13:8 Definition The graph of a quadratic function is a. The is the lowest or highest point of a parabola. The is the vertical line passing through the verte. Graphing Quadratic Functions Using Transformations Step 1: Write the function f a b c as by completing the square in. Step : Graph the function using transformations. 10. Eample: Graph f 4 1 using transformations. Copyright 018 Pearson Education, Inc. 335
Section 7.4 Objective 4: Graph Quadratic Functions of the Form f a b c Part II Eample 8 Video Length 19:53 11. Eample: Graph f 3 6 1 using transformations. 336 Copyright 018 Pearson Education, Inc.
Section 7.4 Objective 5: Find a Quadratic Function from Its Graph Video Length 6:33 If we are given the verte hk,, and one additional point on the graph of a quadratic function, we can find the quadratic function f a b c that results in the given graph. 1. Eample: Find the quadratic function whose graph has a verte of,5 and passes through the point 0, 7. Final answer: Copyright 018 Pearson Education, Inc. 337
Objectives: Section 7.5 Video Guide Graphing Quadratic Functions Using Properties f a b c 1. Graph Quadratic Functions of the Form. Find the Maimum or Minimum Value of a Quadratic Function 3. Model and Solve Optimization Problems Involving Quadratic Functions Section 7.5 Objective 1: Graph Quadratic Functions of the Form f a b c Part I Eample 1 Video Length 1:08 Consider the quadratic function f a b c, a 0 Properties of the Graph of a Quadratic Function f a b c, a 0 Verte = Ais of symmetry; the line Parabola opens if ; the verte is a point. Parabola opens if ; the verte is a point. Consider the function f 3 1 5. 338 Copyright 018 Pearson Education, Inc.
The -intercepts of a Quadratic Function 1. If the discriminant, the graph of f a b c has distinct -intercepts and so will cross the -ais in places.. If the discriminant, the graph of f a b c has -intercept and touches the -ais at its. 3. If the discriminant, the graph of f a b c has -intercept and so will not cross or touch the -ais. y y y (a) (b) (c) 1. Eample: Graph f 3 1 5 using its properties. Copyright 018 Pearson Education, Inc. 339
Section 7.5 Objective 1: Graph Quadratic Functions of the Form f a b c Part II Eample 4 Video Length 5:36. Eample: Graph f 4 7 using its properties. 340 Copyright 018 Pearson Education, Inc.
Section 7.5 Objective : Find the Maimum or Minimum Value of a Quadratic Function Video Length 5:8 The graph of a quadratic function has a verte at. Definition The verte will be the point on the graph if and of f. The verte will be the point on the graph if and of f. Opens f f b a b a will be the will be the Opens 3. Eample: Determine whether the quadratic function minimum value. Find the value. f 3 1 1 has a maimum or Final answer: Copyright 018 Pearson Education, Inc. 341
Section 7.5 Objective 3: Model and Solve Optimization Problems Involving Quadratic Functions Video Length 5:58 4. Eample: The Great Lakes Tour Company offers one-day tours at the rate of $90 per person for each of the first 30 people. For larger groups, each person receives a $0.50 discount. How many people will be required for the tour company to maimize revenue? What is the maimum revenue? Final answer: 34 Copyright 018 Pearson Education, Inc.
Objectives: 1. Solve Quadratic Inequalities. Solve Polynomial Inequalities Section 7.6 Video Guide Polynomial Inequalities Section 7.6 Objective 1: Solve Quadratic Inequalities Part I Eample 1 Video Length 13:43 Definition A is an inequality of the form or or or where a 0. We will go through two different approaches for solving quadratic inequalities. This first approach is a graphical approach and the second is an algebraic approach. Note: The work for the following eample takes up two slides. So make sure you save some room for the graph on the second slide. 1. Eample: Solve the inequality using the graphical method: 3 8 0. Set-builder: Interval notation: Copyright 018 Pearson Education, Inc. 343
Section 7.6 Objective 1: Solve Quadratic Inequalities Part II Eample Video Length 9:13. Eample: Solve the inequality: 3 8 0. Write the steps in words Step 1 Show the steps with math Step Step 3 Step 4 Set-builder: Interval notation: 344 Copyright 018 Pearson Education, Inc.
Section 7.6 Objective 1: Solve Quadratic Inequalities Part III Eample 3 Video Length 10:30 3. Eample: Solve the inequality: 8 Solve graphically: Solve algebraically: Set-builder: Interval notation: Copyright 018 Pearson Education, Inc. 345
Section 7.6 Objective : Solve Polynomial Inequalities Video Length 5:4 Remember, the zeros of a function f are the values that cause the function's value to be 0. If r is a zero, then f r 0. So, the zeros of a function are the same as the -intercepts of the graph of the function. 4. Eample: Solve the inequality: 3 1 4 0. Set-builder: 346 Copyright 018 Pearson Education, Inc.