Ch. 7 Absolute Value and Reciprocal Functions Notes

Transcription

1 First Name: Last Name: Block: Ch. 7 Absolute Value and Reciprocal Functions Notes 7. ABSOLUTE VALUE Ch. 7. HW: p. 364 # 7 odd letters, 9, 3 7. PRE-REQUISITES - GRAPH OF LINEAR FUNCTIONS 4 7. PRE-REQUISITES - GRAPH OF QUADRATIC FUNCTIONS 6 7. ABSOLUTE VALUE FUNCTIONS 8 Ch. 7. HW: p. 375 # all 7.3 ABSOLUTE VALUE EQUATIONS Ch. 7.3 HW: p. 389 # 6 odd letters, # 7,, 4 a, RECIPROCAL FUNCTIONS 6 Ch. 7.4 HW: p. 403 # 9,,, (need a graphing calculator),4, 5, 8 CH. 7 - REVIEW Created by Ms. Lee of 6

2 7. Absolute Value Definition: Absolute value of x, denoted by can be thought of as: A distance or length between zero and x. Eg: Distance between 0 and -5 = -5 = Distance between 0 and 0 = 0 = Therefore, output is always Examples: ) Evaluate the following: a) -7 b) c) 5 3 d) 0 e) 3-3 f) -4 9 g) -(5-7) + 3 h) -+4(3) ) Use absolute value symbols to write an expression for the length of each horizontal or vertical line segment. Determine each length. a) A(, 0) and B(-8, 9) b) A(-, -7) and B (-, 5) Created by Ms. Lee of 6

3 3) On a particular day in Alberta, the temperature was 9 C in the morning. By afternoon, the temperature was raised to 7 C and dropped to 5 C by night. Use absolute value symbols to write an expression for the total change in temperature that day. What is the total change in the temperature for the day? 4) Evaluate. a) b) ( ) c) c) Ch. 7. HW: p. 364 # 7 odd letters, 9, Created by Ms. Lee 3 of 6

4 7. Pre-requisites - Graph of Linear Functions Graph each linear function: ) y = x + 5 ) y = x 4 3 3) y = 4 x + 4) y = 3x 4 5) y = x 6 6) y = x 5 3 Created by Ms. Lee 4 of 6

5 ) y = 5 3 ) y = x 4 3) x = 4) zs y = 3 x + 5) y = x 3 6) y = Created by Ms. Lee 5 of 6

6 7. Pre-requisites - Graph of Quadratic Functions Graph each quadratic function: ) y = 3( x ) 4 ) y = x + 5 3) ( 3) y = x 4) y = ( x + ) ) 3 y = x + 6x 6) y = x + 4x + 3 Created by Ms. Lee 6 of 6

7 Graph each quadratic function in factored form: ) y = ( x )( x + 3) ) y = ( x )( x + ) 3) y = ( x ) 4) y = ( x + 3) 5) y = ( x + )( x + 4) 6) y = ( x )( x + 5) Created by Ms. Lee 7 of 6

8 7. Absolute Value Functions Definition: An absolute value function is a function that involves the absolute value of a variable. A piecewise function is a function composed of two or more separate functions or pieces, each with its own specific domain, that combine to define the overall function. Example: Absolute Value Function of the form, y = ax + b ) Graph = x (use dotted line to represent this function) Graph g ( x) = on the same grid. Define g ( x) = x as a piecewise function. ) Graph = x 3 (use dotted line to represent this function) Graph g ( x) = on the same grid. Define g ( x) = x 3 as a piecewise function. Created by Ms. Lee 8 of 6

9 3) Graph the absolute value function, y = x + 3. Define y = x + 3 as a piecewise function. 4) Graph the absolute value function, = x 4. Define = x 4 as a piecewise function. 5) Given y = f (x) below, graph y =. a) b) Created by Ms. Lee 9 of 6

10 Example: Absolute Value Function of the form, y = ax + bx + c 6) Graph the absolute value function, = x + x + 8. ) Write y = x + x + 8 in factored form. ) Determine the zeros of the function 3) Find the vertex. 4) Sketch y = x + x + 8 (in dotted line) 5) Take the absolute value of y = x + x + 8 and sketch f (x) 7) Express the absolute value function, = x + x + 8 as a piecewise function. Created by Ms. Lee 0 of 6

11 8) Graph the absolute value function, f ( x ) = x 4. Express the absolute value function, 4 f ( x ) = x 4 as a piecewise function. 4 9) Graph the absolute value function, f ( x ) = ( x + ) 4. Express the absolute value function, f ( x ) = ( x + ) 4 as a piecewise function. Ch. 7. HW: p. 375 # all Created by Ms. Lee of 6

12 7.3 Absolute Value Equations ) Solve x = 5 by graphing Solve x = 5 algebraically. Case : Case : ) Solve x + 4 = 3x + 5 by graphing Solve x + 4 = 3x + 5 algebraically. Case : Case : Created by Ms. Lee of 6

13 3) Solve x 5 = 4 x by graphing Solve x 5 = 4 x algebraically. Case : Case : 4) Solve x x = by graphing Solve x x = algebraically. Case : Case : Created by Ms. Lee 3 of 6

14 5) Solve x x = 6) Solve x 5 = x 8x + 5 Created by Ms. Lee 4 of 6

15 7) Solve x 3 = 3x 8) Solve ( x )( x + 3) = 4 9) Determine an absolute value equation in the form ax + b = c given its solutions on the number line. Ch. 7.3 HW: p. 389 # 6 odd letters, # 7,, 4 a, 7 Created by Ms. Lee 5 of 6

16 7.4 Reciprocal Functions Definition: Asymptote: A line that a curve approaches but never crosses (touches) as one of the variables approaches some particular values. Ex: We can just have horizontal asymptotes We can have both horizontal and vertical asymptotes Reciprocal: What is the reciprocal of 4? What is the reciprocal of 5? Given the function, y = f (x). What is the reciprocal of this function? Reciprocal Function: A function defined by where f (x) 0. Functions: y = f (x) y = x Reciprocal Functions: y = Domain of the reciprocal y = x + 5 Created by Ms. Lee 6 of 6

17 y = x 4 y = x + x 6 y = x + 4 How to get horizontal asymptotes: Examine the reciprocal functions, as x approaches ±, the y-value approaches. For any reciprocal functions, y =, would the y-value ever be zero? Therefore, the range of a reciprocal function is:. This means, there will be a asymptote defined by. How to obtain vertical asymptotes: Where there is a non-permissible value, there will be a asymptote. Examples: ) Graph y = = 3x (in red) and its reciprocal (in blue) on the same grid. x f (x) Equation of V.A: Equation of H.A: Created by Ms. Lee 7 of 6

18 ) Graph y = = 3x (in red) and its reciprocal (in blue) on the same grid. x f (x) Equation of V.A: Equation of H.A: 3) Graph y = f ( x ) = x 5 (in red) and its reciprocal (in blue) on the same grid. x f (x) Equation of V.A: Equation of H.A: Created by Ms. Lee 8 of 6

19 4) Graph: y = f (x) = x + 3 (in red) and its reciprocal (in blue) on the same grid x f (x) Equation of V.A: Equation of H.A: 5) State the equation(s) of the vertical asymptote(s) for each function. Reciprocal Functions Non-permissible Values Equations of vertical asymptotes = x + 4 = ( x + )( x 3) g ( x) = (x + ) y = x 9 h ( t) = x + 3 Created by Ms. Lee 9 of 6

20 6) Given the graph of y = f (x), and sketch the graph of the reciprocal function y =. 7) The calculator screen gives a function table for statement for x = ±. =. Explain why there is an ERROR x 4 8) State x-intercept(s) and the y-intercept of each function. a) = x + 3 Created by Ms. Lee 0 of 6

21 b) y = x 7 c) y = x + 3x 0 9) Given the graph of y = f (x), and sketch the graph of the reciprocal function y =. Ch. 7.4 HW: p. 403 # 9,,, (need a graphing calculator),4, 5, 8 Created by Ms. Lee of 6

22 Ch. 7 - Review Multiple Choice Questions:. The zero(es) of y = x occurs when a) x = b) x = c) x = d) x =. The zero(es) of y = x x 4 occurs when a) x = ± 0 b) x = ± 5 c) x = ± d) x = 3) Which of the following is the graph of = ( x + )( x 3). a) b) c) d) 4) Given a linear function, = x, which of the following is the graph of the reciprocal function, y =? a) b) c) d) 5) Determine the equations of the vertical asymptotes of y = x 9. a) x = ± 3 b) x = ± 3 c) x = 3 only d) No V.A. Created by Ms. Lee of 6

23 Written Response: Show all your work clearly for full marks. 6) Evaluate x + 4 x 3 if x =. 7) Write = x as a piece-wise function. The graph of y = f (x) is shown below. [ marks] 8) Write = ( x )( x + 3) as a piece-wise function. [ marks] 9) Write = x + 3x as a piece-wise function. The graph of y = f (x) is shown below. [ marks] 0) Graph each absolute value function, [ marks] a) y = 3x 5 b) y = ( x ) + 3 Created by Ms. Lee 3 of 6

24 Solve x 3 = 7. [3 marks] Domain for Case : Domain for Case : Therefore, solution(s) is/are: ) Solve ( x + )( x ) = x +. [3 marks] Domain for Case : Domain for Case : Therefore, solution(s) is/are: Created by Ms. Lee 4 of 6

25 ) Solve by graphing x + x + 3 = x + 3 [3 marks] Solution(s): 3) Given the graph of y = f (x), sketch the graph of y = few other points. Also draw the asymptote(s) in dotted line(s).. Clearly show invariant points and a [ marks] Created by Ms. Lee 5 of 6

26 4) Solve x x = 6 [3 marks] Domain for Case : Domain for Case : Therefore, solution(s) is/are: Note: Write the solutions in their simplest expressions. Do not round. Created by Ms. Lee 6 of 6