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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 1 Chapter 1 Prerequisites for Calculus Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 1.1 Lines Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review 1. Find the value of y that corresponds to x = 3 in y = - 2 + 4 (x - 3). 2. Find the value of x that corresponds to y = 3 in y = 3 - 2 (x + 1). In Exercises 3 and 4, find the value of m that corresponds to the values of x and y. 3. x = 5, m= y- 3 x- 4 4. x = - 1, y = - 3, m = 2- y 3- x y = 2, Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 4 Quick Review In exercises 5 and 6, determine whether the ordered pair is a solution to the equation. 5. 3 x 4 y 5 6. y 2 x 5 1 a) 2, b) 3, 1 a) 1, 7 b) 2,1 4 In exercises 7 and 8, find the distance between the points. 7. 1,0 and 0,1 1 8. 2,1 and 1, 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 5 Quick Review In Exercises 9 and 10, solve for y in terms of x. 9. 4x- 3 y = 7 10. - 2 x + 5 y = - 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 6 Quick Review Solutions 1. Find the value of y that corresponds to x = 3 in y = - 2 + 4 (x - 3). y= - 2 2. Find the value of x that corresponds to y = 3 in y = 3 - 2 (x + 1). x= - 1 In Exercises 3 and 4, find the value of m that corresponds to the values of x and y. 3. x = 5, m= y- 3 x- 4 m= - 1 4. x = - 1, y = - 3, m = 2- y 3- x m= y = 2, 5 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 7 Quick Review Solutions In exercises 5 and 6, determine whether the ordered pair is a solution to the equation. 5. 3 x 4 y 5 6. y 2 x 5 1 a) 2, b) 3, 1 a) 1, 7 b) 2,1 4 a) yes b) no a) yes b) no In exercises 7 and 8, find the distance between the points. 7. 1,0 and 0,1 1 8. 2,1 and 1, 3 distance = 2 distance = 5/3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 8 Quick Review Solutions In Exercises 9 and 10, solve for y in terms of x. 9. 4 x - 3 y = 7 10. - 2 x + 5 y = - 3 4 7 2 3 y = xy = x3 3 5 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 9 What you’ll learn about… Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications …and why. Linear equations are used extensively in business and economic applications. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 10 Increments If a particle moves from the point (x1 , y1 ) to the point (x2 , y2 ), the increments in its coordinates are x x2 x1 and y y2 y1 The symbols x and D y are read delta x and delta y. The letter D is a Greek capital d for difference. Neither x nor D y denotes multiplication; D x is not delta times x nor is D y delta times y. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 11 Example Increments The coordinate increments from (8, 3) to (-6, 1) are: x 6 8 14, y 1 3 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 12 Slope of a Line Let P1 ( x1 , y1 ) and P2 ( x2 , y2 ) be points on a nonvertical line, L. The slope of L is m= rise y2 - y1 = run x2 - x1 A line that goes uphill as x increases has a positive slope. A line that goes downhill as x increases has a negative slope. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 13 Slope of a Line A horizontal line has slope zero since all of its points have the same y -coordinate, making D y = 0. Dy is undefined. Dx We express this by saying that vertical lines have no slope. For vertical lines, D x = 0 and the ratio Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 14 Parallel and Perpendicular Lines Parallel lines form equal angles with the x-axis. Hence, nonvertical parallel lines have the same slope. m1 = m2 If two nonvertical lines L1 and L2 are perpendicular, their slopes m1 and m2 satisfy m 1 m2 = - 1, so each slope is the negative reciprocal of the other: m1 = - 1 1 , m2 = m2 m1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 15 Equations of Lines The vertical line through the point (a , b) has equation x= a since every x-coordinate on the line has the same value a. Similarly, the horizontal line through (a , b) has equation y = b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 16 Example Equations of Lines Write the equations of the vertical and horizontal lines through the point (- 3,8). x= - 3 and y= 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 17 Point Slope Equation The equation y= m(x - x1 ) + y1 is the point - slope equation of the line through the point (x1 , y1 ) with slope m. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 18 Example: Point Slope Equation Write the point-slope equation for the line through (7, -2) and (-5, 8). 8 - ( - 2) 10 10 5 = = = - 5 - (7) - 12 12 6 We can use this slope with either of the two given points in the point-slope equation. For ( x1 , y1 ) = (7, - 2) we obtain The line's slope is m = 5 (x 6 5 y=- x + 6 5 y=- x + 6 y= - 7) + - 2 35 - 2 6 23 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 19 Equations of Lines The y -coordinate of the point where a non-vertical line intersects the y -axis is the y-intercept of the line. Similarly, the x-coordinate of the point where a non-horizontal line intersects the x-axis is the x-intercept of the line. A line with slope m and y -intercept b passes through (0, b)so y = m ( x - 0) + b, or y = m x + b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 20 Slope-Intercept Equation The equation y=m x + b is the slope - intercept equation of the line with slope m and y-intercept b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 21 General Linear Equation The equation Ax + By = C (A and B not both 0) is a general linear equation in x and y. Although the general linear form helps in the quick identification of lines, the slope-intercept form is the one to enter into a calculator for graphing. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 22 Example Analyzing and Graphing a General Linear Equation Find the slope and y-intercept of the line 2x - 3y = 15. Graph the line. Solve the equation for y to put the equation in slope-intercept form: - 3y = - 2x + 15 - 2 15 y= x + - 3 - 3 2 y= x - 5 3 [-10, 10] by [-10, 10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 23 Example Determining a Function The following table gives values for the linear function f ( x) = mx + b. Determine m and b. x f(x) -1 -1 1 5 3 11 The graph of f is a line. We know from the table that the following points are on the line: (- 1, - 1), (1,5), (3,11) Using the last two points the slope m is: m = 11 - 5 6 = =3 3- 1 2 So f(x) = 3x + b. Because f (1) = 5, we have f (1) = 3(1) + b 5=3 + b b=2 Thus, m = 3, b = 2 and f ( x) = 3x + 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 24 Example Reimbursed Expenses A company reimburses its sales representatives $150 per day for lodging and meals plus $0.34 per mile driven. Write a linear equation giving the daily cost C to the company in terms of x, the number of miles driven. How much does it cost the company if a sales representative drives 137 miles on a given day? Because we know that the relationship is linear, we know that it conforms to the equation C ( x) = .34 x + 150. If a sales representative drives 137 miles, then x = 137. Thus, C (137) = .34(137) + 150 C (137) = 46.58 + 150 C (137) = 196.58 It will cost the company $196.58 for a sales representative to drive 137 miles a day. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 25 1.2 Functions and Graphs Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 - 6, solve for x. 1. 3x - 1 £ 5 x + 3 2. x (x - 2) > 0 3. x- 3 £ 4 4. x- 2 ³ 5 5. x 2 < 16 6. 9 - x 2 ³ 0 In Exercises 7 and 8, describe how the graph of f can be transformed to the graph of g. 2 7. f (x)= x 2 , g (x )= (x + 2) - 3 8. f (x)= x , g (x )= x - 5 + 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 27 Quick Review In Exercises 9 - 12, find all real solutions to the equations. 9. f (x )= x 2 - 5 (a ) f (x)= 4 1 10. f (x)= x (a ) f (x)= - 5 (b) f (x)= - 6 (b) f (x)= 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 28 Quick Review 11. f (x )= x+ 7 (a ) f (x)= 4 12. f (x )= 3 (b) f (x)= 1 x- 1 (a ) f (x)= - 2 (b) f (x)= 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 29 Quick Review Solutions In Exercises 1 - 6, solve for x. 1. 3x - 1 £ 5 x + 3 [- 2, ¥ ) 2. x (x - 2) > 0 (- ¥ , 0)È (2, ¥ ) 3. x- 3 £ 4 [- 1, 7] 4. x- 2 ³ 5 (- ¥ , - 3] È [7, ¥ ) 5. x 2 < 16 (- 4, 4) 6. 9- x2 ³ 0 [- 3,3] In Exercises 7 and 8, describe how the graph of f can be transformed to the graph of g . 2 7. f (x)= x 2 , g (x )= (x + 2) - 3 2 units left and 3 units downward 8. f (x)= x , g (x )= x - 5 + 2 5 units right and 2 units upward Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 30 Quick Review Solutions In Exercises 9 - 12, find all real solutions to the equations. 9. f (x )= x 2 - 5 (a ) f (x)= 4 (a ) - 3,3 (b) f (x)= - 6 (b) no real solution 1 x (a ) f (x)= - 5 (b) f (x)= 0 10. f (x )= (a ) - 1 5 (b) no real solution Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 31 Quick Review Solutions 11. f (x )= x+ 7 (a ) f (x)= 4 (a ) 9 12. f (x )= 3 (b) f (x)= 1 (b) - 6 x- 1 (a ) f (x)= - 2 (a ) - 7 (b) f (x)= 3 (b) 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 32 What you’ll learn about… Functions Domains and Ranges Viewing and Interpreting Graphs Even Functions and Odd functions - Symmetry Functions Defined in Pieces Absolute Value Function Composite Functions …and why Functions and graphs form the basis for understanding mathematics applications. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 33 Functions A rule that assigns to each element in one set a unique element in another set is called a function. A function is like a machine that assigns a unique output to every allowable input. The inputs make up the domain of the function; the outputs make up the range. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 34 Function A function from a set D to a set R is a rule that assigns a unique element in R to each element in D. In this definition, D is the domain of the function and R is a set containing the range. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 35 Function The symbolic way to say " y is a function of x " is y = f (x) which is read as y equals f of x. The notation f (x) gives a way to denote specific values of a function. The value of f at a can be written as f (a), read as " f of a." Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 36 Example Functions Evaluate the function f ( x) = 2 x + 3 when x = 6. f (6) = 2(6) + 3 f (6) = 12 + 3 f (6)= 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 37 Domains and Ranges When we define a function y = f (x ) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of x-values for which the formula gives real y -values the so-called natural domain. If we want to restrict the domain, we must say so. The domain of C (r )= 2p r is restricted by context because the radius, r , must always be positive. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 38 Domains and Ranges The domain of y = 5 x is assumed to be the entire set of real numbers. If we want to restrict the domain of y = 5 x to be only positive values, we must write y = 5 x, x > 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 39 Domains and Ranges The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed or half-open, finite or infinite. The endpoints of an interval make up the interval’s boundary and are called boundary points. The remaining points make up the interval’s interior and are called interior points. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 40 Domains and Ranges Closed intervals contain their boundary points. Open intervals contain no boundary points Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 41 Domains and Ranges Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 42 Graph The points (x, y )in the plane whose coordinates are the input-output pairs of a function y = f (x )make up the function's graph. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 43 Example Finding Domains and Ranges Identify the domain and range and use a grapher to graph the function y = x 2 . Domain: The function gives a real value of y for every value of x so the domain is (- ¥ , ¥ ). Range: Every value of the domain, x, gives a real, positive value of y so the range is [0, ¥ ). y = x2 [-10, 10] by [-5, 15] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 44 Viewing and Interpreting Graphs Graphing with a graphing calculator requires that you develop graph viewing skills. Recognize that the graph is reasonable. See all the important characteristics of the graph. Interpret those characteristics. Recognize grapher failure. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 45 Viewing and Interpreting Graphs Being able to recognize that a graph is reasonable comes with experience. You need to know the basic functions, their graphs, and how changes in their equations affect the graphs. Grapher failure occurs when the graph produced by a grapher is less than precise – or even incorrect – usually due to the limitations of the screen resolution of the grapher. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 46 Example Viewing and Interpreting Graphs Identify the domain and range and use a grapher to graph the function y = x2 - 4 Domain: The function gives a real value of y for each value of x ³ 2 so the domain is (- ¥ , - 2]È [2, ¥ ). Range: Every value of the domain, x, gives a real, positive value of y so the range is [ 0, ¥ ). y= x2 - 4 [-10, 10] by [-10, 10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 47 Even Functions and Odd Functions-Symmetry The graphs of even and odd functions have important symmetry properties. A function y = f ( x)is a even function of x if f (- x) = f (x ) odd function of x if f (- x)= - f (x ) for every x in the function's domain. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 48 Even Functions and Odd Functions-Symmetry The graph of an even function is symmetric about the y-axis. A point (x,y) lies on the graph if and only if the point (-x,y) lies on the graph. The graph of an odd function is symmetric about the origin. A point (x,y) lies on the graph if and only if the point (-x,-y) lies on the graph. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 49 Example Even Functions and Odd Functions-Symmetry Determine whether y = x3 - x is even, odd or neither. y = x3 - x is odd because 3 f (- x)= (- x) - (- x) = - x3 + x = - (x3 - x)= - f (x) y = x3 - x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 50 Example Even Functions and Odd Functions-Symmetry Determine whether y = 2 x + 5 is even, odd or neither. y = 2 x + 5 is neither because f (- x)= 2 (- x)+ 5 = - 2 x + 5 ¹ f ( x) ¹ - f (x ) y = 2x + 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 51 Functions Defined in Pieces While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain. These are called piecewise functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 52 Example Graphing a Piecewise Defined Function Use a grapher to graph the following piecewise function : 2 x 1 x 0 f ( x) 2 x 3 x 0 y = x2 + 3; x > 0 y = 2 x - 1; x £ 0 [-10, 10] by [-10, 10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 53 Absolute Value Functions The absolute value function y = x is defined piecewise by the formula ìïï - x, x < 0 x= í ïïî x, x³ 0 The function is even, and its graph is symmetric about the y-axis Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 54 Composite Functions Suppose that some of the outputs of a function g can be used as inputs of a function f . We can then link g and f to form a new function whose inputs x are inputs of g and whose outputs are the numbers f (g (x )). We say that the function f (g (x )) read ( f of g of x )is the composite of g and f . The usual standard notation for the composite is f o g , which is read " f of g." Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 55 Example Composite Functions Given f ( x) = 2 x - 3 and g (x)= 5x, find f o g. ( f o g ) (x ) = f (g (x)) = f (5 x) = 2 (5 x)- 3 = 10 x - 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 56 1.3 Exponential Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1- 3, evaluate the expression. Round your answers to 3 decimal places. 2 3 1. 5 3. 3- 1.5 2. 3 2 In Exercises 4 - 6, solve the equation. Round your answers to 4 decimal places. 4. x3 = 17 6. x10 = 1.4567 5. x5 = 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 58 Quick Review In Exercises 7 and 8, find the value of investing P dollars for n years with the interest rate r compounded annually. 7. P = $500, r = 4.75%, n = 5 years 8. P = 1000, r = 6.3%, n = 3 years In Exercises 9 and 10, simplify the exponential expression. 9. 2 2 (x y ) 4 3 3 (x y ) - 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2 - 1 æa 3b- 2 ö÷ æa 4 c- 2 ö÷ çç 10. çç 4 ÷ ÷ ÷ çè c ø÷ èç b3 ø÷ ÷ Slide 1- 59 Quick Review Solutions In Exercises 1 - 3, evaluate the expression. Round your answers to 3 decimal places. 2 3 1. 5 3. 3- 1.5 2.924 2. 3 2 4.729 0.192 In Exercises 4 - 6, solve the equation. Round your answers to 4 decimal places. 4. x 3 = 17 6. x10 = 1.4567 ± 1.0383 2.5713 5. x 5 = 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 1.8882 Slide 1- 60 Quick Review Solutions In Exercises 7 and 8, find the value of investing P dollars for n years with the interest rate r compounded annually. 7. P = $500, r = 4.75%, n = 5 years 8. P = 1000, r = 6.3%, n = 3 years $630.58 $1201.16 In Exercises 9 and 10, simplify the exponential expression. 9. 2 2 (x y ) 4 3 3 x ( y) - 3 1 x18 y 5 2 - 1 æa 3b- 2 ö æa 4 c- 2 ö÷ ÷ çç 10. çç 4 ÷ ÷ ÷ çè c ø ÷ èç b3 ø÷ ÷ Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall a2 bc 6 Slide 1- 61 What you’ll learn about… Exponential Growth Exponential Decay Applications The Number e …and why Exponential functions model many growth patterns. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 62 Exponential Function Let a be a positive real number other than 1. The function f ( x) = a x is the exponential function with base a. The domain of f ( x) = a x is (- ¥ , ¥ ) and the range is (0, ¥ ). Compound interest investment and population growth are examples of exponential growth. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 63 Exponential Growth If a 1 the graph of f looks like the graph of y = 2 x in Figure 1.22a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 64 Exponential Growth If 0 a 1 the graph of f looks like the graph of y = 2- x in Figure 1.22b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 65 Rules for Exponents If a > 0 and b > 0, the following hold for all real numbers x and y. x 1. a x ×a y = a x + y 4. a x ×b x = (ab) ax 2. y = a xa x æa ö a 5. çç ÷ ÷ ÷ = bx çè b ø 3. (a x y x y y x ) = (a ) = a xy Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 66 Half-life Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 67 Exponential Growth and Exponential Decay The function y = k ×a x , k > 0, is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 68 Example Exponential Functions Use a grapher to find the zero's of f (x)= 4x - 3. f (x)= 4x - 3 [-5, 5], [-10,10] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 69 The Number e Many natural, physical and economic phenomena are best modeled by an exponential function whose base is the famous number e, which is 2.718281828 to nine decimal places. æ We can define e to be the number that the function f (x)= çç1 + çè x 1 ö÷ ÷ ø x÷ approaches as x approaches infinity. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 70 The Number e The exponential functions y = e x and y = e- x are frequently used as models of exponential growth or decay. Interest compounded continuously uses the model y = P ×e r t , where P is the initial investment, r is the interest rate as a decimal and t is the time in years. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 71 Example The Number e The approximate number of fruit flies in an experimental population after t hours is given by Q (t )= 20 e 0.03 t , t ³ 0. a. Find the initial number of fruit flies in the population. b. How large is the population of fruit flies after 72 hours? c. Use a grapher to graph the function Q. a. To find the initial population, evaluate Q (t ) at t = 0. Q (0)= 20e 0.03(0) = 20e0 = 20 (1)= 20 flies. b. After 72 hours, the population size is Q (72)= 20e 0.03(72) = 20e 2.16 » 173 flies. c. Q(t )= 20e0.03t , t ³ 0 [0,100] by [0,120] in 10’s Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 72 Quick Quiz Sections 1.1 – 1.3 You may use a graphing calculator to solve the following problems. 1. Which of the following gives an equation for the line through (3, - 1) and parallel to the line: y = - 2 x + 1? 1 7 2 2 1 5 B y = x ( ) 2 2 (C) y = - 2 x + 5 (A) y = x + (D) y = - 2 x - 7 (E) y = - 2 x + 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 73 Quick Quiz Sections 1.1 – 1.3 You may use a graphing calculator to solve the following problems. 1. Which of the following gives an equation for the line through (3, - 1) and parallel to the line: y = - 2 x + 1? 7 1 2 2 5 1 (B) y = x 2 2 (C) y = - 2 x + 5 (A) y = x + (D) y = - 2 x - 7 (E) y = - 2 x + 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 74 Quick Quiz Sections 1.1 – 1.3 2. If f (x )= x 2 + 1 and g (x )= 2 x - 1, which of the following gives ( f o g )(2)? (A) 2 (B) 5 (C) 9 (D) 10 (E) 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 75 Quick Quiz Sections 1.1 – 1.3 2. If f (x )= x 2 + 1 and g (x )= 2 x - 1, which of the following gives ( f o g )(2)? (A) 2 (B) 5 (C) 9 (D) 10 (E) 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 76 Quick Quiz Sections 1.1 – 1.3 3. The half-life of a certain radioactive substance is 8 hours. There are 5 grams present initially. Which of the following gives the best approximation when there will be 1 gram remaining? (A) 2 (B) 10 (C) 15 (D) 16 (E) 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 77 Quick Quiz Sections 1.1 – 1.3 3. The half-life of a certain radioactive substance is 8 hours. There are 5 grams present initially. Which of the following gives the best approximation when there will be 1 gram remaining? (A) 2 (B) 10 (C) 15 (D) 16 (E) 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 78 1.4 Parametric Equations Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall What you’ll learn about… Relations Circles Ellipses Lines and Other Curves …and why Parametric equations can be used to obtain graphs of relations and functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 80 Quick Review In Exercises 1 - 3, write an equation for the line. 1. the line through the points (1, 8) and (4, 3) 2. the horizontal line through the point (3, - 4) 3. the vertical line through the point (2, - 3) In Exercises 4 - 6, find the x- and y -intercepts of the graph of the relation. 4. 6. x2 y + =1 9 16 2 y2 = x + 1 5. x2 y 2 =1 16 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 81 Quick Review In Exercises 7 and 8, determine whether the given points lie on the graph of the relation. 7. 2 x2 y + y 2 = 3 (a ) (1, 1) 8. (b) (- 1, - 1) æ1 ö ÷ ç c , 2 ( ) çç ÷ ÷ è2 ø (b) (1, - 3) (c) (- 1, 3) 9 x 2 - 18 x + 4 y 2 = 27 (a ) (1, 3) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 82 Quick Review 9. Solve for t. (a ) 2 x + 3t = - 5 (b) 3 y - 2t = - 1 10. For what values of a is each equation true? (a ) a2 = a (b) a 2 = ± a (c) 4a 2 = 2 a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 83 Quick Review Solutions In Exercises 1 - 3, write an equation for the line. 1. the line through the points (1, 8) and (4, 3) y = - 2. the horizontal line through the point (3, - 4) 3. the vertical line through the point (2, - 3) 5 29 x+ 3 3 y= - 4 x= 2 In Exercises 4 - 6, find the x- and y -intercepts of the graph of the relation. 4. 5. 6. x2 + 9 x2 16 y2 =1 16 y2 =1 9 2 y2 = x + 1 x = - 3, 3; y = - 4, 4 x = - 4, 4; no y -intercepts x = - 1; y= - 1 2 , Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 1 2 Slide 1- 84 Quick Review Solutions In Exercises 7 and 8, determine whether the given points lie on the graph of the relation. 7. 2x2 y + y 2 = 3 (a ) (1, 1) Yes 8. (b) (- 1, - 1) No æ1 ö ç (c) çç , - 2÷÷÷ Yes è2 ø 9 x 2 - 18 x + 4 y 2 = 27 (a ) (1, 3) Yes (b) (1, - 3) Yes Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall (c) (- 1, 3) No Slide 1- 85 Quick Review Solutions 9. Solve for t. (a ) 2 x + 3t = - 5 t= - 2x - 5 3 (b) 3 y - 2t = - 1 t= 3y + 1 2 10. For what values of a is each equation true? (a ) a2 = a (b) a 2 = ± a (c) 4a 2 = 2 a a³ 0 All Reals All Reals Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 86 Relations A relation is a set of ordered pairs (x, y) of real numbers. The graph of a relation is the set of points in a plane that correspond to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter, then we can use the parametric mode of a grapher to obtain a graph of the relation. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 87 Parametric Curve, Parametric Equations If x and y are given as functions x = f (t ), y = g (t ) over an interval of t -values, then the set of points (x, y )= ( f (t ), g (t )) defined by these equations is a parametric curve. The equations are parametric equations of the curve. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 88 Relations The variable t is a parameter for the curve and its domain I is the parameter interval. If I is a closed interval, a £ t £ b, the point ( f (a), g (a)) is the initial point of the curve and the point ( f (b), g (b)) is the terminal point of the curve. When we give parametric equations and a parameter interval for a curve, we say that we have parametrized the curve. A grapher can draw a parametrized curve only over a closed interval, so the portion it draws has endpoints even when the curve being graphed does not. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 89 Example Relations Describe the graph of the relation determined by x = t , y = 1- t 2 . Set x1 = t , y1 = 1- t 2 , and use the parametric mode of the grapher to draw the graph. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 90 Circles In applications, t often denotes time, an angle or the distance a particle has traveled along its path from a starting point. Parametric graphing can be used to simulate the motion of a particle. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 91 Example Circles Describe the graph of the relation determined by x = 3cos t , y = 3sin t , 0 £ t £ 2p . Find the initial points, if any, and indicate the direction in which the curve is traced. Find a Cartesian equation for a curve that contains the parametrized curve. x 2 + y 2 = 9 cos 2 t + 9sin 2 t = 9 (cos 2 t + sin 2 t )= 9 (1)= 9 Thus, x 2 + y 2 = 9 This represents the equation of a circle with radius 3 and center at the origin. As t increases from 0 to 2p the curve is traced in a counter-clockwise direction beginning at the point (3,0). x2 + y 2 = 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 92 Ellipses Parametrizations of ellipses are similar to parametrizations of circles. Recall that the standard form of an ellipse centered at (0, 0) is x2 y 2 + 2 = 1. 2 a b x2 y 2 For x = a cos t and y = a sin t , we have 2 + 2 = 1 a b which is the equation of an ellipse with center at the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 93 Lines and Other Curves Lines, line segments and many other curves can be defined parametrically. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 94 Example Lines and Other Curves If the parametrization of a curve is x = t , y = t + 2, 0 £ t £ 2 graph the curve. Find the initial and terminal points and indicate the direction in which the curve is traced. Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the Cartesian equation is traced by the parametrized curve? x = t , y = t + 2, 0 £ t £ 2 Initial point (0, 2) Terminal point (2, 4) Curve is traced from left to right If x = t and y = t + 2, then by substitution y = x + 2. The graph of the Cartesian equation is a line through (0, 2) with m = 1. The segment of that line from (0, 2) to (2, 4)is traced by the parametrized curve. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 95 1.5 Functions and Logarithms Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 - 4, let f (x)= 3 x - 1, g (x )= x 2 + 1, and evaluate the expression. ( f o g )(1) 3. ( f o g )(x) 1. 2. (g o f )(- 7) 4. (g o f )(x ) In Exercises 5 and 6, choose parametric equations and a parameter interval to represent the function on the interval specified. 1 5. y = , x³ 2 6. y = x, x < - 3 x- 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 97 Quick Review In Exercises 7 - 10, find the points of intersection of the two curves. Round your answers to 2 decimal places. 7. y = 2 x - 3, y = 5 8. y = - 3 x + 5, y = - 3 (a ) y = 2 x , y = 3 10. (a ) y = e- x , y = 4 9. (b) y = 2 x , y = - 1 (b) y = e- x , y = - 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 98 Quick Review Solutions In Exercises 1- 4, let f (x)= 3 x - 1, g (x )= x 2 + 1, and evaluate the expression. 1. 3. ( f o g )(1) 2. (g o f )(- 7) 5 1 ( f o g )(x) x 2 3 4. (g o f )(x ) 2 3 (x - 1) + 1 In Exercises 5 and 6, choose parametric equations and a parameter interval to represent the function on the interval specified. (possible answers are given ) 5. 1 y= , x- 1 x = t, y = x³ 2 1 , t³ 2 t- 1 6. y = x, x < - 3 x = t, y = t, t < - 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 99 Quick Review Solutions In Exercises 7 - 10, find the points of intersection of the two curves. Round your answers to 2 decimal places. (4,5) 7. y = 2 x - 3, y = 5 8. æ8 y = - 3 x + 5, y = - 3 çç , çè3 (a ) y = 2 x , y = 3 10. (a ) y = e- x , y = 4 9. ö 3÷ ÷ ÷ ø (1.58,3) (-1.39, 4) (b) y = 2 x , y = - 1 None (b) y = e- x , y = - 1 None Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 100 What you’ll learn about… One-to-One Functions Inverses Finding Inverses Logarithmic Functions Properties of Logarithms Applications …and why Logarithmic functions are used in many applications including finding time in investment problems. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 101 One-to-One Functions A function is a rule that assigns a single value in its range to each point in its domain. Some functions assign the same output to more than one input. Other functions never output a given value more than once. If each output value of a function is associated with exactly one input value, the function is one-to-one. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 102 One-to-One Functions A function f (x) is one - to - one on a domain D if f (a)¹ f (b) whenever a ¹ b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 103 One-to-One Functions The horizontal line test states that the graph of a one-to-one function y = f (x ) can intersect any horizontal line at most once. If it intersects such a line more than once it assumes the same y -value more than once and is not a one-to-one function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 104 Inverses Since each output of a one-to-one function comes from just one input, a one-to-one function can be reversed to send outputs back to the inputs from which they came. The function defined by reversing a one-to-one function f is the inverse of f. Composing a function with its inverse in either order sends each output back to the input from which it came. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 105 Inverses The symbol for the inverse of f is f - 1 , read "f inverse." The -1 in f - 1 is not an exponent; f - 1 1 (x) does not mean f (x) If ( f o g )(x )= (g o f )(x ), then f and g are inverses of one another; otherwise they are not. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 106 Identity Function The result of composing a function and its inverse in either order is the identity function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 107 Example Inverses Determine via composition if f (x)= x and g (x)= x 2 , x ³ 0 are inverses. ( f o g )(x)= f (x 2 )= x 2 = x = x (g o f )(x)= g ( x ) = 2 ( x) = x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 108 Writing f -1as a Function of x. Solve the equation y = f (x ) for x in terms of y. Interchange x and y. The resulting formula will be y = f - 1 (x ). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 109 Finding Inverses Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 110 Example Finding Inverses Given that y = 4 x - 12 is one-to-one, find its inverse. Graph the function and its inverse. Solve the equation for x in terms of y. 1 x= y + 3 4 Interchange x and y. 1 y= x+ 3 4 f (x)= 4 x - 12 f - 1 (x ) = 1 x+ 3 4 Notice the symmetry about the line y = x [-10,10] by [-15, 8] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 111 Base a Logarithmic Function The base a logarithm function y = log a x is the inverse of the base a exponential function y = a x (a 0, a ¹ 1). The domain of log a x is (0, ¥ ), the range of a x . The range of log a x is (- ¥ , ¥ ), the domain of a x . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 112 Logarithmic Functions Logarithms with base e and base 10 are so important in applications that calculators have special keys for them. They also have their own special notations and names. y = loge x = ln x is called the natural logarithm function. y = log x = log x is often called the common logarithm function. 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 113 Inverse Properties for ax and loga x Base a : a loga x = x, a 1, x> 0 Base e : eln x = x, ln e x = x, x 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 114 Properties of Logarithms For any real numbers x 0 and y 0, Product Rule : log a xy = log a x + log a y Quotient Rule : log a x = log a x - log a y y y Power Rule : log a x = y log a x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 115 Example Properties of Logarithms Solve the following for x. 2 x = 12 2 x = 12 ln 2 x = ln12 Take logarithms of both sides x ln 2 = ln12 PowerRule ln12 2.302585 x= = » 3.32193 ln 2 .693147 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 116 Example Properties of Logarithms Solve the following for x. e x + 5 = 60 e x + 5 = 60 e x = 55 Subtract 5 ln e x = ln 55 Take logarithm of both sides x = ln 55 » 4.007333 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 117 Change of Base Formula log a x = ln x ln a This formula allows us to evaluate log a x for any base a 0, a ¹ 1, and to obtain its graph using the natural logarithm function on our grapher. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 118 Example Population Growth The population P of a city is given by P = 105,300e0.015t where t = 0 represents 1990. According to this model, when will the population reach 150,000? P = 105,300e0.015t , P = 150, 000 150, 000 = 105,300e0.015t 150,000 = e0.015t 105,300 æ150, 000 ÷ ö 0.015t ç ln ç = ln e ÷ çè105,300 ÷ ø Solve for t Take logarithm of both sides 0.353822 = 0.015 t Inverse property 0.353822 t= » 23.588133 years t = 0 is 1990, so 0.015 the population will reach 150,000 in the year 2013. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 119 1.6 Trigonometric Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review In Exercises 1 - 4, convert from radians to degrees or degrees to radians. 1. 3. p 3 - 40° 2. - 2.5 4. 45° In Exercises 5 - 7, solve the equation graphically in the given interval. 5. sin x = 0.6, 0 £ x < 2p 6. cos x = - 0.4, 0 £ x < 2p 7. tan x = 1, - p 3p £ x< 2 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 121 Quick Review 8. Show that f (x)= 2 x 2 - 3 is an even function. Explain why its graph is symmetric about the y -axis. 9. Show that f (x)= x 3 - 3 x is an odd function. Explain why its graph is symmetric about the origin. 10. Give one way to restrict the domain of the function f (x )= x 4 - 2 to make the resulting function one-to-one. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 122 Quick Review Solutions In Exercises 1 - 4, convert from radians to degrees or degrees to radians. 1. p 3 60° 2. - 2.5 - 143.24° 2p p 4. 45° 9 4 In Exercises 5 - 7, solve the equation graphically in the given interval. 3. - 40° 5. sin x = 0.6, 0 £ x < 2p x » 0.6435, 2.4981 6. cos x = - 0.4, 0 £ x < 2p x » 1.9823, 4.3009 7. tan x = 1, - - p 3p £ x< 2 2 x » 0.7854, 3.9270 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 123 Quick Review Solutions 8. Show that f (x)= 2 x 2 - 3 is an even function. explain why its graph is symmetric about the y -axis. 2 f (- x)= 2(- x ) - 3= 2 x 2 - 3= f (x ) The graph is symmetric about the y -axis because if a point (a, b) is on the graph, then so is the point (- a, b). Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 124 Quick Review Solutions 9. Show that f (x)= x 3 - 3 x is an odd function. Explain why its graph is symmetric about the origin. 3 f (- x )= (- x ) - 3(- x )= - x 3 + 3 x = - f (x ) The graph is symmetric about the origin because if a point (a, b)is on the graph, then so is the point (- a, - b). 10. Give one way to restrict the domain of the function f (x )= x 4 - 2 to make the resulting function one-to-one. x³ 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 125 What you’ll learn about… Radian Measure Graphs of Trigonometric Functions Periodicity Even and Odd Trigonometric Functions Transformations of Trigonometric Graphs Inverse Trigonometric Functions …and why Trigonometric functions can be used to model periodic behavior and applications such as musical notes. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 126 Radian Measure The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 127 Radian Measure An angle of measure θ is placed in standard position at the center of circle of radius r, Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 128 Trigonometric Functions of θ The six basic trigonometric functions of q are defined as follows: y r x cosine: cos q = r y tangent: tan q = x sine: sin q = r y r secant: sec q = x x cotangent: cot q = y cosecant: csc q = Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 129 Graphs of Trigonometric Functions When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable (radians) by x instead of θ . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 130 Angle Convention Angle Convention: Use Radians From now on in this book, it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle p p p we mean radians ( which is 60°), not degrees. 3 3 3 When you do calculus, keep your calculator in radian mode. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 131 Periodic Function, Period A function f (x ) is periodic if there is a positive number p such that f (x + p )= f (x ) for every value of x. The smallest value of p is the period of f . The functions cos x, sin x, sec x and csc x are periodic with period 2p . The functions tan x and cot x are periodic with period p . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 132 Even and Odd Trigonometric Functions The graphs of cos x and sec x are even functions because their graphs are symmetric about the y-axis. The graphs of sin x, csc x, tan x and cot x are odd functions. y = cos x y = sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 133 Example Even and Odd Trigonometric Functions Show that csc x is an odd function. 1 1 csc(- x)= = - csc x sin (- x) - sin x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 134 Transformations of Trigonometric Graphs The rules for shifting, stretching, shrinking and reflecting the graph of a function apply to the trigonometric functions. Vertical stretch or shrink Reflection about x-axis Vertical shift y = a f (b (x + c ))+ d Horizontal stretch or shrink Horizontal shift Reflection about the y-axis Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 135 Example Transformations of Trigonometric Graphs Determine the period, domain, range and draw the graph of y = - 2sin (4 x + p ) æ æ p ö÷ ö ç ÷ We can rewrite the function as y = - 2sin ç4 ççx + ÷ ÷÷ ÷ çè çè 4ø ø 2p . In our example b = 4, b 2p p so the period is = . The domain is (- ¥ , ¥ ). 4 2 The graph is a basic sin x curve with an amplitude of 2. Thus, the range is [ - 2, 2]. The period of y = a sin bx is The graph of the function is shown together with the graph of the sin x function. [-5, 5] by [-4,4] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 136 Inverse Trigonometric Functions None of the six basic trigonometric functions graphed in Figure 1.42 is one-to-one. These functions do not have inverses. However, in each case, the domain can be restricted to produce a new function that does have an inverse. The domains and ranges of the inverse trigonometric functions become part of their definitions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 137 Inverse Trigonometric Functions Function Domain y = cos- 1 x - 1£ x£ 1 y = sin- 1 x - 1£ x£ 1 y = tan - 1 x - ¥ < x< ¥ y = sec- 1 x x³ 1 y = csc- 1 x x³ 1 y = cot - 1 x - ¥ < x< ¥ Range 0£ y£ p p p - £ y£ 2 2 p p < y< 2 2 0 £ y £ p, y ¹ - p 2 p p £ y£ ,y¹ 0 2 2 0< y< p Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 138 Inverse Trigonometric Functions The graphs of the six inverse trigonometric functions are shown here. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 139 Example Inverse Trigonometric Functions æ 1÷ ö ç Find the measure of sin ç- ÷ in degrees and in radians. çè 2 ÷ ø - 1 æ 1÷ ö ç Put the calculator in degree mode and enter sin ç- ÷ . çè 2 ÷ ø The calculator returns - 30°. ö - 1æ 1÷ ç Put the calculator in radian mode and enter sin ç- ÷ . çè 2 ÷ ø - 1 The calculator returns - .52359877556 radians. p This is the same as radians. 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 140 Quick Quiz Sections 1.4 – 1.6 You should solve the following problems without using a graphing caluclator. 1. Which of the following is the domain of f ( x) log x 3 ? 2 (A) , (B) ,3 (C) 3, (D) [3, ) (E) ( ,3] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 141 Quick Quiz Sections 1.4 – 1.6 You should solve the following problems without using a graphing caluclator. 1. Which of the following is the domain of f ( x) log x 3 ? 2 (A) , (B) ,3 (C) 3, (D) [3, ) (E) ( ,3] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 142 Quick Quiz Sections 1.4 – 1.6 2. Which of the following is the range of f ( x) 5cos x 3? (A) , (B) 2,4 (C) 8, 2 (D) 2,8 2 8 (E) , 5 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 143 Quick Quiz Sections 1.4 – 1.6 2. Which of the following is the range of f ( x) 5cos x 3? (A) , (B) 2,4 (C) 8, 2 (D) 2,8 2 8 (E) , 5 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 144 Quick Quiz Sections 1.4 – 1.6 3. Which of the following gives the solution of tan x -1 in x (A) (B) (C) 3 ? 2 4 4 3 3 (D) 4 5 (E) 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 145 Quick Quiz Sections 1.4 – 1.6 3. Which of the following gives the solution of tan x -1 in x (A) (B) (C) 3 ? 2 4 4 3 3 (D) 4 5 (E) 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 146 Chapter Test In Exercises 1 and 2, write an equation for the specified line. 1. through (4, - 12) and parallel to 4 x + 3 y = 12 2. the line y = f (x ) where f has the following values: -2 2 x f(x) 4 2 4 1 1 5 3. Determine whether the graph of the function y = x is symmetric about the y -axis, the origin or neither. 4. x4 + 1 Determine whether the function y = 3 is even, odd or neither. x - 2x Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 147 Chapter Test In Exercises 5 and 6, find the (a ) domain and (b) range, and (c)graph the function. 5. y = 2sin (3x + p )- 1 6. y = ln (x - 3)+ 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 148 Chapter Test 7. Write a piecewise formula for the function. 1 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 149 Chapter Test 8. x = 5cos t , y = 2sin t , for a curve. 0 £ t £ 2p is a parametrization (a ) Graph the curve. Identify the initial and terminal points, if any. Indicate the direction in which the curve is traced. (b) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 150 Chapter Test 9. Give one parametrization for the line segment with endpoints (- 2, 5) and (4,3). 10. Given f (x)= 2 - 3x, (a ) find f - 1 and show that ( f ○ f - 1 )(x)= ( f - 1 ○ f )(x ) (b) graph f and f - 1 in the same viewing window Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 151 Chapter Test Solutions In Exercises 1 and 2, write an equation for the specified line. 4 20 1. through (4, - 12) and parallel to 4 x + 3 y = 12 y = - x3 3 2. the line y = f (x ) where f has the following values: x -2 2 f(x) 4 3. 2 4 1 x+ 3 2 1 1 5 Determine whether the graph of the function y = x is symmetric about the y -axis, the origin or neither. 4. y= - Origin x4 + 1 Determine whether the function y = 3 is even, odd x - 2x or neither. Odd Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 152 Chapter Test Solutions In Exercises 5 and 6, find the (a ) domain and (b) range, and (c)graph the function. 5. y = 2sin (3x + p )- 1 (a ) All Reals (b) [- 3, 1] [-π, π] by [-5, 5] 6. y = ln (x - 3)+ 1 (a )(3, ¥ ) (b) All Reals [- 2, 10] by [- 2, 5] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 153 Chapter Test Solutions 7. Write a piecewise formula for the function. 1 1 2 ìïï 1- x, 0 £ x < 1 f (x)= í ïïî 2 - x, 1£ x £ 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 154 Chapter Test Solutions 8. x = 5cos t , y = 2sin t , for a curve. 0 £ t £ 2p is a parametrization (a ) Graph the curve. Identify the initial and terminal points, if any. Indicate the direction in which the curve is traced. Initial Point (5, 0) Terminal Point (5, 0) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 155 Chapter Test Solutions 8. (b) Find a Cartesian equation for a curve that contains 2 2 æx ÷ ö æy ö÷ ç the parametrized curve. ç ÷ + çç ÷ =1 çè 5 ÷ ø èç 2 ø÷ What portion of the graph of the Cartesian equation is traced by the parametrized curve? All Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 156 Chapter Test Solutions 9. Give one parametrization for the line segment with endpoints (- 2, 5) and (4,3). (A possible answer) x = - 2 + 6t , y = 5 - 2t , 0 £ t£ 1 10. Given f (x )= 2 - 3 x, (a ) find f - 1 and show that ( f ○ f - 1 )(x)= ( f - 1 ○ f )(x ) 2- x 3 æ2 - x ö æ2 - x ö - 1 ÷ ÷ ç f ( f (x ))= f ç = 2 - 3çç ÷= 2 - (2 - x)= x ÷ ÷ çè 3 ÷ ç ø è 3 ø (a ) f - 1 (x)= f - 1 ( f (x)) = f - 1 2 - (2 - 3 x) 3 x = =x (2 - 3x)= 3 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 157 Chapter Test Solutions 10. (b) graph f and f - 1 in the same viewing window f - 1 (x )= 2- x 3 f (x)= 2 - 3x [-5, 5] by [-5, 5] Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 1- 158