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CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS C. SPECIAL FUNCTIONS 1. Absolute Value Function (AVF) Ex 1: An AVF is a function whose output is determined by the rules of absolute value. All parts of the function enclosed in the absolute value bars must return a positive value. Notice how the function is defined in three parts. The rule says that if the input x is positive the result stays the same and if the input is zero then the output is zero. Notice when x is negative (x < 0), the input is multiplied by a negative one to insure a positive output. x ; x0 f (x) x 0 ; x 0 x ; x 0 The graph of absolute value is a piece wise defined function. The graphs of y x , y 0 and y x are sketched on a single grid. By restricting each graph you obtain the parent graph that defines the y x . D x , and R x 0, Ex 2: Write each absolute value function without using absolute value notation. (1.) f (x) x a You have to write three separate rules. So you need a rule for when the expression in the bars is a positive number, zero and a negative number. The expression (x + a) is zero when x = –a, positive when x > –a and negative when x < –a. When you replace x with a number less than a negative a, the value of (x + a) is negative. Since you want a rule that will return a positive, all you do is multiply the output by a (–1). ; x a xa f (x) x a 0 ; x a 1(x a) ; x a (2.) f (x) x a Notice that the a is not in the absolute value bar. Therefore it has no effect on the restrictions. The restrictions are determined solely by what is inside the bars. ; x0 xa f (x) x a a ; x0 (1x) a ; x 0 Revised 09-01-2007 gfmartin Page 1 of 6 CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS Ex 3: There are various techniques for graphing Absolute Value Functions. The method really depends on the expression between the bars (inside the absolute value). (1.) Translations and/or Dilation: This works well with any AVF where the rule inside the bars is a linear expression and there are no variables outside the bars. Sketch f (x) x 3 2 You can graph this function by starting with the parent graph and then moving the parent graph 3 units horizontally in the positive direction and then 2 units vertically in the positive direction. (2.) The Definition of the Absolute Value Function: This will work for any AVF. Sketch f (x) x 3 2 Start by rewriting the function as three separate functions with the proper restrictions. x 1 if f (x) x 3 2 2 if x 5 if x3 x3 x3 Now graph the piece-wise defined function, paying special attention to the restrictions. Start by first plotting the point (3,2) and then sketching in the graph of the line y x 1 to the right of the point and the graph of the line y x 5 to the left of the point. (3.) Graphing Calculator: Just type the equation into the grapher using the absolute value command. The ABS is found by pressing the MATH key and choosing the NUM menu. Revised 09-01-2007 gfmartin Page 2 of 6 CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS Ex 4: Sketch the graph of each function. Verify your result with a graphing calculator. (1.) Ex 5: f (x) x 1 (2.) f (x) 1 x (3.) f (x) x 3 Sketch the graph of the function f (x) x 1 3 2x This graph cannot be generated using translation/dilation. The first step in graphing this function is to rewrite it with the proper restrictions. (1.) Rewrite the function as three separate functions with the proper restrictions. Remember that only what is inside of the absolute value bars effects the restrictions. _______________________ ; x _________ f (x) x 1 3 2x _______________________ ; x _________ _______________________ ; x _________ 2. (2.) Sketch the graph of f. Start by plotting the focal point x = and then using the rule for x > and x <. (3.) Verify your graph by using the ABS command. The Greatest Integer Function (GIF) Ex 6: The Greatest Integer Rule: The “greatest integer” of a real number x is the integer n such that n ≤ x ≤ n+1. n x n+1 Translated into English this means take the first integer to the left of the real number x on the number line. The notation [[ x ]] = n can be read as the greatest integer of x is n. (1.) [[ 5.25 ]] = ______ (2.) [[ –2.5 ]] = ______ Revised 09-01-2007 (3.) [[ 1.99 ]] = ______ gfmartin Page 3 of 6 CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS Ex 7: The Greatest Integer function f(x) = [[ x ]] is also know as the step function. Every input will produce an output that is an integer. The domain may be any real number and the range must be an integer. The parent graph is given below. Note the holes in the graph. Ex 8: To sketch a greatest integer function with a linear argument (ax + b) you must know the step size. The step size is determined by taking the reciprocal of the slope. This will tell you how long each step is and will also give you the scale you should use with your graph. Sketch: f (x) ß3x 1® The reciprocal of the slope is 1/3. Therefore your scale on the x-axis should be in 1/3 increments. At each increment find the corresponding y value. Plot the point. Draw in the steps. Decide if the endpoints are included or excluded. You can verify your graph by using your grapher. Press the MATH key, choose NUM and then choose int(. This command can be used to graph the GIF. Set your x-scale to 1/3, your step size and y-scale to 1. Note that the graph does not display the open or closed circles at the endpoints of each step. You must do that. Also notice that the graph does look like a series of steps. You can fix that my going to MODE and choosing DOT in the menu. (1.) Sketch f(x) = [[ x+1 ]] (2.) Revised 09-01-2007 Sketch f(x) = [[ 3–2x ]] gfmartin Page 4 of 6 CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS 3. Periodic Functions – Trigonometric Functions Ex 9: The function y f( x) C Asin B(x D) y -ax is Period = P = 2š B y = C +A A un its y=C A un its D + P eriod D 1 y = C– A x- ax is Ex 10: Sketch the graph of y f ( x) 5 3cos 4 x 4 . Verify you result with a graphing calculator. Be sure you are in radian mode. 4. Exponential Functions and Logarithmic Functions Ex 11: Exponential functions and Logarithmic functions are inverses of each other. Therefore their parent graphs are reflections of each other over the line y = x. Below are the parent graphs of y ex and y ln x . Remember that e is a number (a transcendental number) that has a value of ≈ 2.71828 . . .. So the functions are called base e exponentials and base e logarithms. Revised 09-01-2007 gfmartin Page 5 of 6 CALCULUS AB UNIT TWO: FUNCTIONS – SPECIAL FUNCTIONS Ex 12: Conversion between base a exponentials and logarithms. y ax x log a y x y3 4 2x x log 3 (y) x log 2 (4) 9 a3 3 log a (9) Ex 13: Properties of Logarithms (1.) loge (A) ln A loge 4 ln 4 (2.) loga (AB) loga (A) loga (B) loga 4x loga 4 loga x (3.) A log a log a (A) log a (B) B log a x 3 log a x 3 log a x 2 x2 (4.) log a An n loga (A) loga 9 loga 32 2 loga 3 (5.) log a b ln a ln b log 3 7 ln 7 ln 3 Revised 09-01-2007 gfmartin Page 6 of 6