Foundations of Math II Unit 5: Solving Equations
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1 Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics
2 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following two lines: Y = 2x 4 Y = 10 What is the point of intersection? Explain what this point represents. 2
3 5.1 Solving Linear Equations Using Graphing, Tables, and Algebraic Properties Lesson (Lesson from 3
4 4
5 5.1 Practice Solving Linear Equations Using Graphing, Tables, and Algebraic Properties 5
6 5.2 Warm Up Solving Quadratic Equations by Graphing and Tables Using a table of values, graph the function f(x)= x 2 + 2x 8 Identify the following key characteristics: Domain: Range: Vertex: Axis of Symmetry: x-intercept(s): y-intercept: Increasing Interval: Decreasing Interval: 6
7 5.2 Lesson Handout - Solving Quadratic Equations by Graphing and Tables Today we will be using quadratic functions to assist us in solving quadratic equations. Definition: A quadratic equation is an equation that can be written in the standard form, where a, b, and c are real numbers and a 0. The solutions of a quadratic equation are called its roots or zeros. Notice the similarity between a quadratic equation and a quadratic function. Quadratic Function: y = ax 2 + bx + c Quadratic Equation: 0 = ax 2 + bx + c SOLVING A QUADRATIC EQUATION BY USING ITS RELATED QUADRATIC FUNCTION Graphically Using a Table Graph the related quadratic Create a table for the function and find the x-intercepts quadratic function and find (because this is where y = 0). the x values for which y = 0. Exercise #1: The graph of the function y = x 2 3x 4 is given. According to the graph, the roots of y the equation x 2 3x 4 = 0 are Exercise #2: A table of values is given below for the function y = x 2 2x 15. Use the table to determine the values of x for which x 2 2x 15 = 0. x y
8 Exercise #3: Find the zeros of the function y = x 2 + 3x 54 numerically by using a table of values. Create the table with your graphing calculator. Exercise #4: The graph of a particular function of the form y = ax 2 + bx + c, where a, b, and c are real numbers, is shown below. Use the graph to answer the following questions. (a) Is the numerical value of a positive or negative? Justify. (b) State the numerical value of c. How do you know? (c) State all solutions to each equation below: (i) ax 2 + bx + c = 0. (ii) ax 2 + bx + c = 3. Exercise #5: Using the accompanying grids, sketch graphs of functions of the form y = ax 2 + bx + c that satisfy the given criteria. (a) ax 2 + bx + c = 0 has two (b) ax 2 + bx + c = 0 has exactly (c) ax 2 + bx + c = 0 has roots; a > 0 one root; a < 0 no roots; a > 0 8
9 Fact: A quadratic equation can have,, or real solutions Exercise #6: Determine the roots for each quadratic equation given below by using your graphing calculator. (a) x 2 + x 6 = 0 (b) x 2 6x 9 = 0 (c) x 2 + 4x + 6 = 0 Exercise #7: Which of the following functions has two real zeros? (1) y = x 2 + 7x +14 (3) y = x 2 5x + 6 (2) y = x 2 10 x + 25 (4) y = x Exercise #8: If one x-intercept of the graph of a quadratic function is 4 and the axis of symmetry has equation x = 3, then what is the other x-intercept? Fact: A quadratic equation can have,, or real solutions Exercise #9: Which table below illustrates a quadratic function with a maximum value of zero? 9
10 5.2 Practice Solving Quadratic Equations by Graphing and Tables Solving Quadratic Equations by Graphing The solutions of a quadratic equation are called the roots of the equation. You can find the real number roots by finding the x-intercepts or zeros of the related quadratic function. Quadratic equations can have two distinct real roots, one distinct root, or no real roots. These roots can be found by graphing the equation to see where the parabola crosses the x-axis. Describe the real roots of the quadratic equations whose related functions are graphed below. a) b) c) State the real roots of each quadratic equation whose related function is graphed below m 16 = 0 6. g 2 + 4g 5 = 0 Solve each equation by graphing. If integral roots cannot be found, state the consecutive integers between which the roots lie. 9. n = 4n 10. w 2 = -2w 11. v 2 = -6v t 2 = x 2 + 2x 3 = 0 5. m k 2 8k = h 2 3 = The real roots of a quadratic equation correspond to the? of the graph of the related function. A. x-intercepts B. y-intercepts C. vertex D. maximum 10
11 5.2 Homework - Solving Quadratic Equations by Graphing and Tables 1. The graph of y = x 2 6x + 8 is shown. The roots of the equation x 2 6x + 8 = 0 are 2. Which of the following graphs illustrates a quadratic function that has no real zeros? (A) (B) (C) (D) (4) x 3. Determine the zeros for each quadratic function given below by using your graphing calculator. (a) y = x 2 + 3x 10 (b) y = x 2 5x 6 (c) y = x 2 + 8x +16 (d) y = x (e) y = x 2 2x 4 (f) y = x x 2 11
12 4. A table of values is given below for the function y = 27 6x x 2. Use the table to determine the values of x for which 27 6x x 2 = 0. x y How many real roots does x 2 + 7x + 6 = 0 have? 6. If the two x-intercepts of the graph of a quadratic function are 3 and 9, then the equation of the axis of symmetry is 7. If one x-intercept of the graph of a quadratic function is 4 and the axis of symmetry has an equation of x = 7, then what is the other x-intercept? 8. The graph of a particular function of the form y = ax 2 + bx + c, where a, b, and c are real numbers, is shown below. Use the graph to solve each of the following equations. (a) ax 2 + bx + c = 0 (b) ax 2 + bx + c = 1 (c) ax 2 + bx + c = 3 12
13 5.3 Warm- Up Greatest Common Factor Find the greatest common factor for each of the following: 1. 12, m 2 n, 21m 3 n 3. 80x 3, 30yx n, 140m 2, 80m 2 13
14 5.3 Lesson Handout Solving Quadratic Equations by Factoring Example: 3n 2-6n-45 Review of Factoring Using the X-Box Method STEPS: Example: 1. Factor out GCF. Complete the remaining steps on the remaining trinomial. 2. Draw a big X. 3. Multiply the lead coefficient and the constant together. Put the answer in the top of the X. 4. Put the middle term number of the trinomial in the bottom of the X. 14
15 5. Find two numbers that will multiply to give you the top number and add to give you the bottom number. 6. Put those two numbers in the sides of the X. 7. Rewrite the original trinomial by removing the middle term and replacing the term as a sum by using the two numbers you found in step 5. (Don t forget the variables!) 8. Put the polynomial from step 6 into a box. 15
16 9. Factor the GCF from each row and each column of the box. Place these numbers on the top and the side of the box. 10. Read your answer from the sides of the box. (Don t forget the GCF!) Now try these: 1. f(x) = 12x 2 +17x+6 answer: 2. f(x) = 4x 2-100y 2 answer: 16
17 3. f(x) = x 2 +14x+24 answer: 4. f(x) = 6x 3 +15x 2-9x answer: 5. f(x) = 2h 2-3h-18 answer: 17
18 6. f(x) = a 2 +18a+81 answer: 7. f(x) = 32x 2-80x+50 answer: 8. f(x) = 6y 2-5y-6 answer: 18
19 Solving Quadratics by Factoring Steps: Example: 1. Quadratic Equation: n 2 = -18 9n 2. Put equation into standard form and set equal to zero: 3. Factor the equation: 4. Set each factor equal to zero: 5. Solve each smaller equation: 19
20 Now try these: Solve each by factoring 2 a. (x + 3)(x 7) = 0 b. x 3x c. 2t 17t 45 3t 5 d. 9t 12t e. 3x 6 x 10 f. 2w 10w 23w w
21 5.3 Practice Solving Quadratic Equations by Factoring 21
22 5.3 Homework Solving Quadratic Equations by Factoring Solve each quadratic by factoring. 1. x(x-7) = 0 2. p 2-19p + 70 = 0 3. (2z + 1) 2 = 0 4. y 2 = 3y 5. c 2-9 = 0 6. h = -9h 7. -f - 6 = -f x - 6 = -4x 2 9. r 3-2r 2-15r = z 2 - z = v 2 + 3v = x x =
23 5.4 Warm-Up Factoring Factor completely: 1. x 2 7x k 2 + 9k 3. 7x 2 45x 28 Explain how you know to just factor or to solve a quadractic. Explain the steps for factoring a quadratic. 23
24 5.4 Lesson Handout Solving Quadratics Using the Quadratic Formula 24
25 25
26 5.4 Practice Solving Quadratics Using the Quadratic Formula Solve each equation using the quadratic formula: b x b 2 4ac 2a 1. x 2 7x + 10 = 0 2. x 2 14x + 45 = x x 10.5 = y y 11.7 = k k = 0 6. x 2 11x x 2 + 8x y 2 8y 3 = 0 9. x x = z z = x 2 = x 12. x 2 10x + 35 = 7x 35 26
27 5.4 Homework Solving Quadratics Using the Quadratic Formula 27
28 5.5 Warm Up - Solving Exponential Functions Using Tables and Graphs 1. Use a table of values to graph the following two functions: a. f(x) = 3 2 x x f(x) b. g(x) = 2(½) x x f(x) 2. For each function, identify the initial value, the y- intercept, and whether the function is a growth or decay. 28
29 5.5 Practice: Solving Exponential Functions Using Tables and Graphs 29
30
31 5.5 Homework: Solving Exponential Functions Using Tables and Graphs 31
32 32
33 5. Suppose that when you are 15 years old, a magic genie gives you the choice of investing $10,000 at a rate of 7% or $5000 at a rate of 12%. Either choice will be compounded annually. The money will be yours when you are 65 years old. Which investment would be the best? Justify your answer
34 5.6 Warm-Up Solving Equations Solve. Explain your reasoning behind why you solved each problem the way you did. 1. ( 4k + 5 )( k + 1 ) = 0 2. n 2 10n + 22 = r 2 14r = r 2 44r = r 5. 15a 2 3a = 3 7a 34
35 5.6 Practice-Introduction to Common Logs and Properties of Logs 35
36 36
37 5.6 Homework-Introduction to Common Logs and Properties of Logs 37
38 38
39 5.7 Warm-Up Log Properties Condense each log expression. 1. 8log 5 a + 2log 5 b 2. 3logx 5logy 3. logz + logx + logy log log
40 5.7 Practice-Solving Exponential Equations with Logs 40
41 5.7 Homework-Solving Exponential Equations with Logs Solve the following: 41
42 5.8 Warm-Up Graphing Radical Functions Graph each function using your calculator. Sketch a picture of what you see. Use a standard viewing window. Make sure you plot key points on your sketch (y-intercept and x-intercept(s)) 1. y = x y = x y = 2 x
43 5.8 Practice-Solving Radical Functions By Graphing and Algebraic Properties 43
44 5.8 Homework-Solving Radical Functions By Graphing and Algebraic Properties 44
45 5.9 Warm-Up Adding and Subtracting Fractions and Solving Proportions Add or subtract the following fractions a + 7 x a b Solve each proportion x+1 = 5 x = x+1 x
46 5.9 Practice- Solving Rational Equations 46
47 5.9 Homework Solving Rational Equations 47
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