The online MML Assignment due dates are uniform for all classes and are not subject to change. In-class homework due dates are set by your TA.

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1 The online MML Assignment due dates are uniform for all classes and are not subject to change. In-class homework due dates are set by your TA. MML HW2.1,2.2 Sun Jun 16 HW Book pg. 92: 22, 44, 60, 70 MML HW2.3 Tues Jun 18 MML HW2.4 Thurs Jun 20 HW Book pg. 99: 38, 56 pg. 111: 30, 58, 90 pg. 122: 52, 58, 70 pg. 132: 8, 14, 36 pg. 144: 24abc, 37, 78, 88 MML HW2.5, 2.6a,3.2 Sun Jun 23 MML HW 3.3, Quiz 2 Tues Jun 25 HW Book pg. 174: 24, 32, 60, 64 pg. 189: 36, 68, 72, 96 EXAM Friday June 28 Section Function Transformations Objectives: Basic Functions Worksheet 1. Use transformations (shifts, stretches, shrinks, and/or reflections) to write equations of functions ns need to be memorized. from a description You should and be able fromto asketch graph. them accurately and quickly. 2. Use transformations (shifts, stretches, shrinks, and/or reflections) to graph functions. 2 n y x 3. Use shifts, stretches, shrinks, 2. and/or Squaring reflections Function to write y equations x of functions from a graph. Transformations represent one way to create new functions from a known function. Exploring Transformations Complete the following by sketching the graphs in your notes. The graph of the basic function f(x) = x is shown below. 3. Square Root Function y x n x y y x 5. Cube Root Function y! 3 x! Use your graphing calculator to graph the following functions and compare the result to the basic graph. Each is a transformation of the original. Compare both the shape of the graph (describe the relationship) and list some ordered pairs for each graph using the TABLE feature in your graphing calculator. 6. Absolute Value Function y x! 1. f(x) = x f(x) = x 1 1 ction 1 y! 8. Logarithmic Function y ln x

2 3. f(x) = x f(x) = x 2 5. f(x) = x 6. f(x) = x 7. f(x) = 2x 8. f(x) = 3 x 2

3 1. Vertical and Horizontal Shifts (Section 2.2) f(x) + c where c is a real number, shifts the graph of f(x) vertically. f(x + c) where c is a real number, shifts the graph of f(x) horizontally. Vertical shifts change the function outputs (y-values). The x-values do not change. Horizontal shifts change the inputs (x-values) to the function. The y-values do not change. Task 1: Given an equation write a description y 1 = x Describe the transformation of f(x) = x. Identify the domain/range for both. Task 2: Given a description write an equation Write the function that shifts y = x 2 two units left and one unit up. Task 3: Given a graph write an equation Write the equation for the graph below. Assume each grid mark is a single unit.! Task 4: Equation graph. Sketch the graph of y = f(x) = x 2 1 How does the transformation affect the domain and range? 3

4 2. Stretches, Shrinks, & Reflections (Section 2.3) cf(x), where c is a real number, stretches (or shrinks) the graph of f(x) vertically. f(cx), where c is a real number, shrinks (or stretches) the graph of f(x) horizontally. f(x) reflects the graph of f(x) vertically about the x-axis. f( x) reflects the graph of f(x) horizontally about the y-axis. Vertical stretches, shrinks, reflections change the function outputs (y-values). The x-intercepts do not change. Horizontal stretches, shrinks, reflections change the x-intercepts, but not the y-intercept. Task 5: Graph(generic) & symbolic transformation new graph Given f(x): f(x) + 2 f(x + 2) f(x 1) 3 2f(x) f(x) f( x) 4

5 3. Combining Transformations of Graphs Summary Let a, b, c, d be positive or negative constants. Consider f(x): cf(bx + a) + d Horizontal Modifications to f(x): a: Horizontal shift b: Horizontal stretch, shrink, reflection Vertical Modifications to f(x): c: Vertical stretch, shrink, reflection d: Vertical shift Order of Transformations: When multiple transformations are involved, use the following sequence of operations to perform changes on the appropriate coordinates to create the transformed graph of the function. (a) Horizontal shift: Subtract a from each x-coordinate (b) Horizontal stretch, shrink, reflection: Divide each x-coordinate by b (c) Vertical stretch, shrink, reflection: Multiply each y-coordinate by c (d) Vertical shift: Add d to each y-coordinate. Example 1 Use transformations to sketch the graph of: y = 1 x The equation has the form: f(x) = cf(bx + a) + d where a = b = c = d = 5

6 Example 2: Use the graph of f(x) below and the rules for combining multiple transformations to sketch the graph of 2f(x + 1) 3!!! The graph of a function, f(x), is shown. Use transformations to sketch the graph of: (a) f(x + 1) 3 (b) -2f(x 2) + 1 (c)! f(x) - 2 6

7 Section Absolute Value Functions Objectives: 1. Solve absolute value equations analytically and graphically. 2. Solve absolute value inequalities analytically and graphically. 1. Definitions: Geometric: The absolute value of a number or expression is the magnitude of the quantity, regardless of direction. Informally, absolute value is the distance from zero. Symbolic (algebraic) { : x, if x 0 f(x) = x = x, if x < 0 f(x) { EX 1, 2 f(x), if f(x) 0 f(x) = f(x), if f(x) < 0 Given the graph of f(x) below, sketch the graph of f(x) 2. Solving equations - analytically and graphically. EX 4 Graphic solutions should be clearly labeled, with the solution to the equation identified. Analytically Graphically (using intersect method) x 3 = 7 y 1 = x 3 & y 2 = 7 Answer (as a solution set): { } 7

8 3. Solving inequalities - analytically and graphically. EX 5 Graphic solutions to an inequality should be clearly labeled, with the solution in terms of x identified. Analytically Graphically 2x + 1 < 5 Analytically x Graphically 4. Special cases. Match each example with the appropriate solution. EX 7 i) 3x + 2 = 2 a) (, 0) b) (0, ) ii) x 7 > 1 c) (, ) d) 0 iii) 1 2x < 5 e) 0, 4 3 f) 3x + 1 = 2x 7 EX 8, 9 Answer (as a solution set): { } 8

9 Section Piecewise-defined Functions Objectives: 1. Graph piecewise-defined functions. Interpret the graph of piecewise-defined functions. 2. Write a formula for a piecewise-defined graph. Piecewise-defined functions are combinations of two or more functions, each defined on non-intersecting restricted domains. Study pg , watch the videos for Examples 1, 2, 3 from the Multimedia textbook in MML, then complete #1 & #2 below. 1. Get information from the graph of a piecewise-defined function. EX 1 x + 3 if 3 x < -1 f(x) = 5 if 1 x 1 x if 1 < x < 9 (a) What is the domain? (b) What is the range? (c) Evaluate f( 1), f(1), f(3) 2. Graph a piecewise-defined function. EX 2 4 if x 0 f(x) = x 2 if 0 < x 2 2x 6 if x > 2 9

10 Math 101 Section 2.5 Piecewise-Defined Functions a. Find f(2) and f(8). b. Find x when f(x) = 1. c. State the domain. d. State the range Graph the piecewise function. 2x + 7 g( x) = 1 3 x if x if 4 x 0 if0 < x < According to the Walden Pond State College course catalog, Tuition for lower division (0-54 credits) undergraduate students who are Walden residents taking anywhere from 12 to 16 credit hours is the same, a total of $3,620 per semester. Tuition is $315 per credit hour for fewer than 12 credits and for each credit over 16. a. Define a piecewise function T that gives the tuition cost as a function of credit hours taken. Assume that the maximum number of credit hours that a lower division undergraduate student may take is 20. b. State the domain and range of T. 10

11 Section Combinations of Functions Objectives: 1. Use tables, graphs, and function expressions to form new functions by combining arithmetically. f(x + h) f(x) 2. Determine the difference quotient for a given function and simplify completely. h 1. Create new functions by combining known functions using arithmetic operations. EX 1, 2, 3 Addition: Subtraction: Multiplication: Division: (f + g)(x) = f(x) + g(x) (f g)(x) = f(x) g(x) (f g)(x) = f(x) g(x) ( ) f (x) = f(x) g g(x), g(x) 0 The domain of f + g, f g, or f g include all the real numbers in the intersection of the domains of the individual functions. The domain of f is the intersection of the domains of f and g for which g g(x) 0. f(x) = 3x 1 g(x) = x 2 4 h(x) = x + 2 p(x) = x + 3 (a) Evaluate: (f + g)(6); Find (f + g)(x) (b) Find (f h)(x) (c) Find (g f)(x). Is this the same as (f g)(x)? (d) Find ( g h) (x) (e) Find (h p)(x) (f) Evaluate: (h p)(1); (h p)( 4) 11

12 2. Using tables and graphs to form combinations. f g x f(x) g(x) Find: (f + g)(1) (g f)(3) (f g)( 1)! ( ) f (3) g 3. Special case: Simplifying the Difference Quotient. EX 4 f(x + h) f(x) h Example 1: Find and simplify the difference quotient for the function f given by f(x) = 3x 2 + 2x Example 2: Find and simplify the difference quotient for the function f given by f(x) = 4 x 2 Section Complex Numbers Objectives: 1. Simplify square roots of negative values. 2. Perform operations (+,, ) with complex numbers. 3. Simplify powers of i Vocabulary: Complex number Real number Imaginary unit Complex conjugate 12

13 Complex numbers are used in the remainder of the course. Practice the operations in MML. If you need to review the concepts, study Section 3.1. The following notes summarize the main points. Some functions have zeros that are not real numbers. In order to find the zeros of such functions, we must consider the complex-number system. We know that the square root of a negative number is not a real number. In particular, 1 is not a real number because there is no real number x such that x 2 = 1. This means that equations like x 2 = 1 or x = 0 do not have real-number solutions, and functions like f(x) = x do not have real zeros. To deal with these situations, we will define a non-real number to be the solution to the equation x = 0. The number i is defined such that i = 1 and i 2 = 1 A complex number is a number of the form a + bi, where a and b are real numbers. The number a is said to be the real part of a + bi and the number b is said to be the imaginary part of a + bi. Sometimes the complex number bi is called a pure imaginary number. The following are examples of complex numbers: For the complex number 3 5i, 3 is the real part and 5 is the imaginary part. The Complex number system includes all real numbers. 2 is a complex number because it can be expressed as 2 + 0i. The real part is -2, imaginary part 0. 4i is a pure imaginary number. Real part 0, imaginary part 4 All numbers (except possibly the positive integers) are creations of the human mind - the numbers -1 and 2 as well as the number i. We study complex numbers because they complete the study of solutions to equations. Practical uses for complex numbers are found in quantum mechanics and electrical engineering, although the particular study is beyond the scope of this course. 13

14 Write in terms of i. Simplify any radicals Powers of i i 1 = i i 2 = 1 i 3 = i 2 i = i i 4 = i 2 i 2 = 1 1 = 1 Use the Powers of i to simplify. 3. i (i) 42 Operations with complex numbers result in other complex numbers. Operation Description Example Addition To add complex numbers, add the real parts and the imaginary parts (6 i) + (3 + 7i) Subtraction Multiplication To subtract complex numbers, subtract the real parts and the imaginary parts (8 + 14i) ( 3 6i) Multiply complex numbers like binomials, using i 2 = 1 (6 2i)(3 + 5i) (7 + 4i)(7 4i) In the last example, 7 + 4i and 7 4i are examples of complex conjugates. conjugates will always be a real number. The product of complex 14

15 Division with complex numbers is accomplished by multiplying the numerator and denominator of the quotient by the complex conjugate of the denominator. The result is similar to rationalizing the denominator with radical expressions. Example 1: i = i i 3(5 + 11i) = i = i Example 2: 3 + 2i 1 i = 3 + 2i 1 i 1 + i 1 + i = 1 + 5i = i Complex number operations can also be performed with most graphing calculators. For the TI-83/84 calculators, select a + bi under the MODE menu. The i is a 2nd function of the decimal point on the bottom row of the calculator. 15

16 Section Quadratic Functions & Graphs Objectives: 1. Transform a quadratic equation into the form f(x) = a(x h) 2 + k by completing the square. 2. Analyze the graph of a quadratic function. 3. Graph quadratic functions by using the vertex and symmetry. 4. Use the vertex formula to find the extremum (maximum or minimum) of a quadratic function. Vocabulary: Perfect square binomial Vertex Let a, b, c, be real numbers with a 0. A quadratic function is given as: General Form f(x) = ax 2 + bx + c Standard Form f(x) = a(x h) 2 + k 1. Convert between forms: Express in general form: f(x) = (2x 1)(x + 3) f(x) = 2(x + 3) 2 2 Express in standard form (complete the square to find a perfect square binomial): f(x) = x 2 + 4x + 3 f(x) = 3x 2 6x f(x) = x x

17 2. Characteristics of the graph of a quadratic: EX 2, 3, 4 The graph of a quadratic functions is called a. If the leading coefficient, a, is positive [a > 0], the graph opens. If the leading coefficient, a, is negative [a < 0], the graph opens. The turning point of the graph is called the vertex. For the graph of f(x) = a(x h) 2 + k, the vertex is (h, k). Extrema: The vertex is a if a is positive and the vertex is a if a is negative. The vertical line x = h that intersects the vertex is called the. Intercepts y int: f(0) = c x int: Solve f(x) = 0 (Section 3.3) Increasing, decreasing intervals 3. Completing the square (see example above) is one way to find the vertex. There is also a vertex formula. If the function is in general form, the vertex is ( b 2a, f( b 2a )) and the axis of symmetry is x = b 2a. Note: See page 169 for the development of the vertex formula from the general form of a quadratic equation. Example: P (x) = 3x 2 + 4x + 1 The vertex for P (x) is 4. Analyzing graphs. Find the equation of the quadratic function (parabola) graphed below. a) What is the vertex? b) Identify the x-intercepts. c) Write an equation for the parabola. d) Identify the axis of symmetry. e) What is the domain? range? f) Over what interval is the graph increasing? decreasing? 17

18 5. Graphing quadratic functions (analysis) g(x) = 2x 2 + 6x 3 Determine if the graph opens up or down. Determine the vertex (h, k) Find the y-intercept. Plot the vertex and at least 2 additional points, (symmetry) and connect with a smooth curve. Additional analysis: (a) What is the largest interval over which the graph is increasing? decreasing? (b) What is the domain? the range? 6. Applications: Writing the equation of a parabola given the vertex and one point on the graph. Write the equation of the parabola with vertex at (8, 3) passing through (10, 5). Note (8, 3) (h, k) and (10, 5) (x, f(x)) Use these values to calculate a. f(x) = a (x h) 2 + k Try: Vertex (5,6); through (1, -6) Projectile motion. Position equation: s(t) = 16t 2 + v 0 t + s 0 EX 6 The equation s(t) = 16t t describes the height of a projectile fired vertically upward after t seconds. Graph this function in the window [0, 60] by [-1000, 11000]. (a) From what height was the projectile fired? (b) After how many seconds will it reach its maximum height? (c) What is the maximum height? (d) Between what two times will the projectile be more than 5000 feet above the ground? (e) How long will the projectile be in the air? 18

19 Section Solving Quadratic Equations & Inequalities Objectives: 1. Solve quadratic equations: zero-product property, square root property, quadratic formula. 2. Determine the nature and number of solutions to a quadratic equation by examining the discriminant. 3. Write the equation for a quadratic function given the vertex and one point. 4. Use a sign test to solve quadratic inequalities. 5. Use the graph of the function to solve quadratic inequalities. Vocabulary: zero of a function perfect square of a binomial zero-product property quadratic formula discriminant critical values maximum (minimum) of a quadratic Let a, b, c, be real numbers with a 0. A quadratic function is given as: f(x) = ax 2 + bx + c or f(x) = a(x h) 2 + k 1. Solving quadratic equations find the real or complex zeros of the quadratic function. That is, solve: f(x) = 0 Zero-product Property. EX 1, 2 If a b = 0, then either a = 0, b = 0, or both. Examples: (2x 3)(3x + 2) = 0 3y(y 1) = 0 x x = 12 x 2 49 = 0 Square root Property. EX 3 Examples: x 2 = 49 x 2 = 4 (4x 3) 2 = 36 (1 z) 2 = 8 19

20 Quadratic Formula. EX 4, 5 For a quadratic equation in the form ax 2 + bx + c = 0 with a 0, x = b ± b 2 4ac 2a Examples: Clearly identify the values for a, b, and c. x 2 + 5x + 7 = 0 (x 3)(x + 5) = 7 In the quadratic formula, the expression b 2 4ac is called the. The value of b 2 4ac gives information about the zeros of the quadratic function. b 2 4ac > 0 b 2 4ac = 0 b 2 4ac < 0 2. Find a quadratic function that has the following zeros: 4 and and Solving quadratic inequalities Solve the corresponding quadratic equation to identify the zeros. Analytical solution: Sign test. EX 6, 7 Solving a quadratic inequality analytically means finding the set of x-values (inputs) that will produce (output) values for the quadratic that are greater than or less than zero. Ex. x 2 + 4x 12 0 First set the equation equal to zero to find its critical values (its zeros). x 2 + 4x 12 = 0 20

21 Create a table using the factors and intervals formed by the critical values to determine where the equation is greater than or equal to zero. Select a test value from each interval to represent all results from that interval. [The sign test shown here is slightly different from the one shown in the text.] Intervals: (, 6) ( 6, 2) (2, ) Test value: (x + 6) ( ) (+) (+) (x 2) ( ) ( ) (+) Product: (+) ( ) (+) Solution to? Ans: (, 6] [2, ) Graphical solution: When the quadratic expression is compared to 0, (a) >, solution is x-interval(s) that make the graph lie above the x-axis; (b) <, solution is the x-interval(s) that make the graph lie below the x-axis. Examine the graph of y 1 = x 2 + 4x 12 to verify the analytical solution. 4. Solve x 2 + 5x 6. 21

22 Section Modeling with Quadratic Functions Objectives: 1. Solve applications using quadratic models. Key Concepts: 1. Perimeter of a rectangle: P = 2l + 2w 2. Area of a rectangle = length width A = lw 3. Volume of a rectangular prism (box) = length width height V = lwh 4. Finding an extreme (max/min) for a parabola involves finding the vertex. 1. Classic Furniture Concepts has determined that when building x hundred wooden chairs the average cost per chair can be modeled by C(x) = 0.1x 2 0.7x where C(x) is in hundreds of dollars. How many chairs should the company build in order to minimize the average cost per chair? 2. EX 1 Suppose you want to enclose a rectangular region bordering on a river with fencing as shown in the diagram. Let x represent the length of each of the parallel pieces of fencing and assume there is only 600 feet of fencing available. (a) What is the length of the remaining piece in terms of x? (b) Determine a function A that represents the Area of the enclosed region? (c) Are there any restrictions on x? (d) What dimensions would give a total area of 22,500 sq. ft.? (e) What is the maximum area that can be enclosed? 22

23 3. Finding the volume of a box EX 2 Alternate example: A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2-inch squares from each corner and folding up the sides. Let x represent the width of the original piece of cardboard. Page 1 of 1 (a) What are the dimensions of the bottom of the box? (b) What are the restrictions on the value of x? (c) Determine a function V that represents the volume of the box in terms of x? (d) For what dimensions of the bottom of the box will the volume be 320 cubic inches? file://c:\users\clyde\documents\box01.gif 5/28/