Unit 2 - Functions and Their Graphs PDF

PRE-CALCULUS Unit 2 Created by: ALL THINGS ALGEBRA® Thank you SO MUCH for purchasing this product! I hope you found this resource useful in your classroom! Please consider leaving feedback in my TpT store or email me at [email protected] with any questions or comments. You can also fi...

PRE-CALCULUS Unit 2 Created by: ALL THINGS ALGEBRA® Thank you SO MUCH for purchasing this product! I hope you found this resource useful in your classroom! Please consider leaving feedback in my TpT store or email me at [email protected] with any questions or comments. You can also find me here: TERMS OF USE © 2012-2018 Gina Wilson (All Things Algebra ®) LICENSING TERMS: By purchasing this product, the purchaser receives a limited individual license to reproduce the product for use within their classroom. This license is not intended for use by organizations or multiple users, including but not limited to school districts, schools, or multiple teachers within a grade level. This license is non-transferable, meaning it can not be transferred from one teacher to another. If other teachers in your department would like to use this product, additional licenses can be purchased from my TpT store. 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Thank you for respecting my work! Unit 2 – Functions & Their Graphs: Sample Unit Outline TOPIC HOMEWORK Functions Review DAY 1 • • • • Representing Relations (Tables, Mappings, Graphs, Equations) Domain and Range Relations vs. Functions Function Notation & Evaluating Functions HW #1 DAY 2 Intercepts; Zeros; Critical Points (Extrema & Points of Inflection); Increasing & Decreasing Intervals HW #2 DAY 3 Quiz 2-1 None DAY 4 Continuity; End Behavior HW #3 DAY 5 Tests for Symmetry; Even and Odd Functions HW #4 DAY 6 Average Rate of Change HW #5 DAY 7 Quiz 2-2 None DAY 8 Parent Functions & Transformations HW #6 DAY 9 Graphing Functions HW #7 DAY 10 Piecewise Functions HW #8 DAY 11 Quiz 2-3 None DAY 12 Function Operations & Compositions of Functions HW #9 DAY 13 Inverse Relations & Functions DAY 14 Quiz 2-4 DAY 15 Unit 2 Review DAY 16 UNIT 2 TEST HW #10 None Study For Test None Note: The parent functions graphic organizer on Day 8 includes linear, absolute value, quadratic, cubic, square root, cube root, reciprocal, exponential, logarithmic, greatest integer, sine, and cosine. However, only these functions are included when it comes to graphing functions: linear, absolute value, quadratic, cubic, square root, cube root, reciprocal, and greatest integer. This unit focuses primarily on these functions. Other functions will be the focus of later units. © Gina W ilson (All Things Algebra ®, LLC), 2017 Name: ___________________________________________________ ________________ Topic: ___________________________________________________ ________________ Main Ideas/Questions Notes/Examples relations Date: _________________________________ _________________________________ Class: _ _________________________________ _________________________________ _ Example: Representing Relations x y DOMAIN RANGE Directions: Give the domain and range of each relation. Examples 2. 1. 3. x -2 -1 0 1 2 {(-1, -1), (7, 4), (3, 0), (-5, -1)} D= D= D= R= R= R= y 0 1 4 Directions: Give the domain and range of each relation in set notation. 4. 5. 6. D= D= D= R= R= R= © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Give the domain and range of each relation in interval notation. 7. 8. 9. D= D= D= R= R= R= 10. 11. 12. D= D= D= R= R= R= A function is a special type of relation, in which each element functions of the ________________ is paired with ________________ ________ element of the ______________. Directions: Determine whether the relation is a function. 13. 14. x y -3 -1 2 5 -4 -2 1 4 x y -9 0 4 -2 -1 1 5 15. {(-7, 2), (-5, 2), (-2, 8), (0, 8)} If a vertical line intercepts a relation at more than one point, then the relation is not a function. vertical line test Directions: Determine whether the relation is a function. 16. 17. 18. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________________________ ________________ Topic: ___________________________________________________ ________________ Main Ideas/Questions Notes/Examples EXPLICIT EQUATIONS IMPLICIT EQUATIONS EXAMPLES Date: _________________________________ _________________________________ Class: _ _________________________________ _________________________________ _ Examples: Examples: Directions: Write each equation explicitly in terms of x. Then, indicate whether the equation is a function. 1. x  4 y  20 2. x2  y2  25 3. ( y  1)2  x  3 4. 1 x 2 3 y 7 6. xy  5 y  1  2 x 5. x  y  1  8 Explicit functions are frequently written using function notation: FUNCTION NOTATION evaluating functions Equation Function Notation This is read as “_______________________” and interpreted as the value of the function f at x. Directions: Evaluate each function for the given value. 7. f ( x)  x2  6x  23; f (5) 8. h( x)  7  x2 ; h(9) © Gina Wilson (All Things Algebra®, LLC), 2017 9. p( x)  x 1 ; p(4) 2 2 x  4 x  13 10. g ( x)  x3  3 x2  7x  4; g (2) 1 x  14 ; p( 10) 2 11. q( x)  6  2 x ; q(15) 12. p( x)  13. f ( x)  5  (3) ; f (4) 14. a( x)   2 x  7  1; a(17) x 15. f ( x)  x3  x2 ; f (3 y 2 ) 17. h( x)  x2  x  1; h(2a  7) 3 16. g ( x)  3 x  10 ; g ( 5c) x5 18. k ( x)  4 x3  7; k (m  1) Directions: Using the graph to the left, find each function value. 19. f (1) 20. f ( 2) 21. f (5) 22. f (0) © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 1: Relations and Functions ** This is a 2-page document! ** Directions: Give the domain and range of each relation and determine whether it is a function. 1. 2. 3. x -5 -3 0 4 y -2 { ( −3, −9),( −2, −1), ( −2, −6),(−1, −9) } -1 7 D= D= D= R= R= R= Function? Function? Function? Directions: Give the domain and range of graphs 4, 5, and 6 in set notation and graphs 7, 8, and 9 in interval notation. Determine whether each relation is a function. 4. 5. 6. D= D= D= R= R= R= Function? Function? Function? 7. 8. 9. D= D= D= R= R= R= Function? Function? Function? © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Write each equation explicitly in terms of x. Then, indicate whether the equation is a function. 3 11. y 2 − x2 + 1 = 50 10. x − y = −12 2 12. 3 xy + x = y − 6 13. y 2 + 6 y + 9 = x + 8 Directions: Evaluate each function for the given value. 3 2 14. f ( x) = − x + 2 x − 11; f (−3) 16. p( x) = x2 − 4 ; p(−8) 2x + 1 18. q( x) = 2 x4 − 3 x2 ; q(−2 x2 ) 20. k ( x) = −2 x − 15 ; k (−6w) x+9 15. g ( x) = 13 − x 2 ; g (11) 17. h( x) = 3 ⋅ 9x ; h(−2) 19. j( x) = − x2 + 3 x + 10; j(3 x − 2) 21. t ( x) = x3 − 2 x; t ( x + 4) © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • Intercepts are points where a graph intersects the _____ or _____- _________. x- and yIntercepts • An x-intercept occurs where ___________. x-intercept x-intercept • A y-intercept occurs where ___________. • A function can have 0+ x-intercepts, y-intercept but at most one y-intercept. Examples Directions: Identify the x- and y-intercepts of the functions graphed below. 1. 2. 3. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: 4. 5. 6. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: Zeros Find Zeros & the y-intercept Algebraically • To find the zeros, set ____________________ and solve for ______. • To find the y-intercept, find ______________. © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Find the zero(s) and y-intercept of each function algebraically. Examples 7. f ( x) = − x − 5 + 3 8. f ( x) = 2 x 2 − 7 x − 4 zero(s): zero(s): y-intercept: y-intercept: 9. f ( x) = x 4 − 5 x 2 + 4 10. f ( x) = x 3 − 6 x 2 − 7 x zero(s): zero(s): y-intercept: y-intercept: 11. f ( x) = 3 x 3 + 13 x 2 − 10 x 12. f ( x) = zero(s): zero(s): y-intercept: y-intercept: x +1 − 2 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • Critical points are points where the graph maximum changes its __________________ or ______________. CRITICAL POINTS • Extrema are points where the graph changes _____________________. The function has a minimum or maximum value (relative or absolute) at these points. point of inflection • A point of inflection is where the graph changes its ______________, but not its direction. minimum A point with the greatest value compared to points around it. (also called a local maximum) Relative & Absolute EXTREMA absolute maximum relative maximum The point with the greatest value over the entire domain of the function. A point with the least value compared to points around it. (also called a local minimum) relative minimum The point with the least value over the entire domain of the function. EXAMPLES absolute minimum Directions: Give the coordinates and classify the extrema for the graph of each function. Use your graphing calculator to approximate if needed. 2 1. f ( x) = 2 x − 16 x + 27 4 2 2. f ( x) = − x + 4 x − 2 x − 4 © Gina Wilson (All Things Algebra®, LLC), 2017 3 2 5 3. f ( x) = − x − 5 x − 3 x + 4 3 4. f ( x) = x − 3 x + 3 Moving from left to right, a function may increase, decrease, or remain constant on certain intervals. INCREASING & DECREASING Behavior EXAMPLES For example, the function to the right is • increasing on the interval ____________. • constant on the interval ____________. • decreasing on the interval ____________. Directions: Give the intervals on which each function increases or decreases. Use your graphing calculator to approximate if necessary. 5. 6. 7. Increasing Interval(s): Decreasing Interval(s): 3 2 8. f ( x) = − x + 4 x − 2 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 4 3 2 9. f ( x) = x + x − 4 x + 1 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 5 10. f ( x) = x − 2 x Increasing Interval(s): 3 Decreasing Interval(s): © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 2: Intercepts, Zeros, Extrema ** This is a 2-page document! ** Directions: Identify the x- and y-intercepts of the functions graphed below. 1. 2. 3. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: 4. 5. 6. x-intercept(s): x-intercept(s): x-intercept(s): y-intercept: y-intercept: y-intercept: Directions: Find the zero(s) and y-intercept of each function algebraically. 8. f ( x) = x 3 − 10 x 2 + 24 x 7. f ( x) = 2 x − 5 − 1 zero(s): y-intercept: 4 2 9. f ( x) = 4 x − 17 x + 4 zero(s): zero(s): 10. f ( x) = y-intercept: zero(s): y-intercept: 1 x−3 2 y-intercept: © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Give the coordinates and classify the extrema for the graph of each function. Use your graphing calculator to approximate if needed. 11. f ( x) = 2 x 2 + 4 x − 5 12. f ( x) = x 3 + 6 x 2 + 9 x + 5 13. f ( x) = − x 5 + 4 x 3 − 2 14. f ( x) = x 4 − 4 x 3 + 9 x Directions: Give the intervals on which each function increases or decreases. Use your graphing calculator to approximate if necessary. 15. 16. 17. Increasing Interval(s): Decreasing Interval(s): 18. f ( x) = − x 5 + 3 x 3 + 1 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 19. f ( x) = − 2 x − 5 + 6 Increasing Interval(s): Decreasing Interval(s): Increasing Interval(s): Decreasing Interval(s): 20. f ( x) = x 3 + 2 x 2 − 3 x − 4 Increasing Interval(s): Decreasing Interval(s): © Gina Wilson (All Things Algebra®, LLC), 2017 Name: _________________________________________________ Pre-Calculus Date: ______________________________Per: ________ Unit 2: Functions & Their Graphs Quiz 2-1: Characteristics of Functions (Part 1) Directions: Give the domain and range of each relation using set notation. Then, indicate whether the relation shown by the graph is a function. 1. 2. 3. D = ________________________ D = ________________________ D = ________________________ R = ________________________ R = ________________________ R = ________________________ Function? _________ Function? _________ Function? _________ Directions: Write each equation explicitly in terms of x. Write your answer in the box, then indicate whether the equation is a function. 1 2 4. 6 x − 3 y = 21 5. y − x + 2 x = −5 6. 8 x 2 + 4 y 2 = 12 3 Function? __________ Function? __________ Function? __________ Directions: Given f(x) = x3 + 2x and g(x) = -x2 – 5x + 12, find each function value. Write your answers to the right. 7. f (−3) 8. g (9) 7. __________________ 8. __________________ 9. __________________ 9. g ( −5a) 10. f (4c − 1) 10. __________________ © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Identify the x-intercepts (zeros) and y-intercept of each function graphed below. 11. 12. x-intercept(s): x-intercept(s): y-intercept: y-intercept: Directions: Find the zero(s) and y-intercept of each function algebraically. 4 2 13. f ( x) = x − 10 x + 9 zero(s): 14. f ( x) = x+4 −3 y-intercept: zero(s): y-intercept: Directions: Using the graphs below, classify each point as a point of inflection, relative minimum, absolute minimum, relative maximum, or absolute maximum. Graph 1 F Graph 2 C D 15. Point A: _________________________________ 16. Point B: __________________________________ E B 17. Point E: __________________________________ A 18. Point F: __________________________________ Directions: Give the increasing and decreasing intervals of each function. Use your graphing calculator to approximate if necessary. 19. f ( x) = −2 x 2 − 8 x − 1 20. f ( x) = x 3 − 9 x 2 + 24 x − 19 Increasing Interval(s): Increasing Interval(s): Decreasing Interval(s): Decreasing Interval(s): © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • A graph is said to be ______________________________ if there are no ______________ or _________________ in Continuity the graph. • If there are breaks or holes in the graphs, it is considered to be ______________________________. Types of Discontinuity a When f(x) increases or decreases infinitely as x approaches x = a from the left or the right. Examples a a When the f(x) jumps from one point to another at x = a. When f(x) is continuous everywhere else except for a hole at x = a. Directions: Determine if the function shown on the graph is continuous. If not, identify the type and location of discontinuity. 1.\ 2. 3. 4. 5. 6. © Gina Wilson (All Things Algebra®, LLC), 2017 End Behavior As x moves towards positive infinity and negative infinity, end behavior describes what is happening to the value of f(x). Describe the end behavior of the graph to the right: As x → ∞ , f ( x) → ________ As x → −∞ , f ( x) → ________ Examples Directions: Describe the end behavior of each graph. 7. 8. 9. As x moves towards positive infinity and negative infinity, it is possible for f(x) to be approaching a certain value. Graphs Approaching a Certain Value Describe the end behavior of the graph to the right: As x → ∞ , f ( x) → ________ As x → −∞ , f ( x) → ________ Examples Directions: Describe the end behavior of each graph. 10. 11. 12. 13. 14. 15. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 3: Continuity & End Behavior ** This is a 2-page document! ** Directions: Determine if the function shown on the graph is continuous. If not, identify the type and location of discontinuity. 1. 2. 3. 4. 5. 6. 7. 8. 9. © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Describe the end behavior of each graph. 10. 11. 12. 13. 14. 15. 16. 17. 18. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions TYPES OF SYMMETRY Notes/Examples The graph can be folded along a line so that the two halves match. The graph remains unchanged when rotated 180° about a point. y y x y x x The _____-___________ and _____-___________ are common lines of symmetry. The _______________ is a common point of symmetry. The following graphical and algebraic tests can be used to determine if the graph of a relation is symmetric to the x-axis, y-axis, and/or origin. TESTS FOR SYMMETRY origin y-axis x-axis Graphical Test Algebraic Test For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is results in equivalent equations. also on the graph. For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is results in equivalent equations. also on the graph. For every point (x, y) on the Replacing ______ with ______ graph, the point _________ is AND ______ with ______ results also on the graph. in equivalent equations. Directions: Use the graph to determine if the relation is symmetrical to the Examples x-axis, y-axis, and/or origin. Confirm your answer algebraically. 1. y = x 4 − 4 x 2



x-axis y-axis origin none 2. xy = 6



x-axis y-axis origin none © Gina Wilson (All Things Algebra®, LLC), 2017 3. 4( x + 3)2 + y 2 = 16



5. y =



4. y 2 − x 2 = 1



x-axis y-axis origin none 3 x −1 x-axis y-axis origin none 6. y = − x − 2



x-axis y-axis origin none x-axis y-axis origin none Algebraic Check: A function is __________ if it is symmetric with EVEN AND ODD respect to the _____-__________. Functions A function is __________ if it is symmetric with Algebraic Check: respect to the ________________. Examples Directions: Determine algebraically if the function is even, odd, or neither. If even or odd, describe the symmetry. 3 2 7. f ( x) = x − x 8. f ( x) = − x + 6 9. 11. f ( x) = x 3 − x 2 − 2 x f ( x) = −8 x x2 − 4 10. f ( x) = 12. f ( x) = x + 1 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 4: Tests for Symmetry; Even and Odd Functions ** This is a 2-page document! ** Directions: Use the graph to determine if the relation is symmetrical to the x-axis, y-axis, and/or origin. Confirm your answer algebraically. 2. xy  4 1 1. y  x 3 4  x-axis  x-axis  y-axis  origin  none  y-axis  origin  none 3. y  2 x  5     x-axis y-axis origin none 5. y 2  x  4     x-axis y-axis origin none 4. y  3 2 x  3     x-axis y-axis origin none 6. x2  y2  36     x-axis y-axis origin none © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Determine algebraically if the function is even, odd, or neither. If even or odd, describe the symmetry. x 7. f ( x) = −3 x4 + 7x2 − 1 8. f ( x) = 7 9. f ( x) = x − 4 10. f ( x) = 5 x − 6 11. f ( x) = −4 x3 + 9x 12. f ( x) = x2 + 2 x − 5 13. f ( x) = 2 x − 9 14. f ( x) = 3 5x © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples Directions: Find the slope of the line that passes between the given points. Slope Review Average Rate of Change 1. 2. 3. 4. (-9, 3) and (-5, -7) 5. (2, -1) and (-7, -4) 6. (-13, -5) and (-1, 3) For linear functions, the slope is constant between each pair of points. For nonlinear functions, the slope changes between different pairs of points. Because of this, we find the average rate of change. • The average rate of change of a function between two points is the slope of the line that passes through these points. • This is called a _________________ __________. • Formula for the average rate of change of a function on the interval [a, b]: a Examples b Directions: Given the function graphed below, find the average rate of change on each interval. 7. [-2, 1] 8. [0, 4] © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Find the average rate of change of each function on the given interval. Round your answer to the nearest hundredth if necessary. 2 9. f ( x) = − x + 6 x − 14; [−1, 2] 3 2 11. f ( x) = x − 2 x − 9 x + 1; [4, 7] 13. f ( x) = Applications x−4 ; [2, 8] x 2 10. f ( x) = 2 x + 28 x + 93; [−9, − 7] 4 2 12. f ( x) = − x + 3 x − 17; [−2, 5] 14. f ( x) = x + 5; [−1, 7] 15. A company’s total cost C(x) in dollars to manufacture x units is given by C(x) = 0.05x2 + 40x + 1200. Find the average rate of change of total cost for the first 200 units manufactured. 16. A ball is thrown straight upward so that its height h(t), in feet, is given by the equation h(t) = -16t2 + 80t, where t is time in seconds. Find the average rate of change in the height of the ball from when it reaches its maximum height until when it reaches the ground. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 5: Average Rate of Change ** This is a 2-page document! ** Directions: Given the function below, find the average rate of change on each interval. 1. [-5, -2] 2. [-3, 0] 3. [-9, 7] 4. [-2, 0] Directions: Find the average rate of change of each function on the given interval. Round your answer to the nearest hundredth if necessary. 5. f ( x) = x2 + 7 x − 11; [−8, − 5] 6. f ( x) = 7 − x ; [−9, 6] 7. f ( x) = x ; [−1, 5] x+3 8. f ( x) = − 1 2 x + 3 x + 1; [8,14] 2 © Gina Wilson (All Things Algebra®, LLC), 2017 9. f ( x) = x3 + 2 x2 − 5 x + 3; [1, 5] 10. f ( x) = x2 ; [−1,10] x+2 11. f ( x) = 1 4 x − x 3 ; [−2, 2] 4 12. f ( x) = 3 4 x − 1; [0, 7] 13. f ( x) = 2x − 1 ; [−3, 8] x+4 14. f ( x) = x3 + 5 x 2 − 2; [−9, − 5] 15. The average low temperature by month in Nashville is represented by the function f ( x) = −1.4 x 2 + 19x + 1.7 , where x is the month. Find the average rate of change from March to August. 16. A rocket was launched into the air from a podium 6 feet off the ground. The rocket path is represented by the equation h(t ) = −16t 2 + 120t + 6 , where h(t) represents the height, in feet, and t is the time, in seconds. Find the average rate of change from the initial launch to the maximum height. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: _________________________________________________ Pre-Calculus Date: ______________________________Per: ________ Unit 2: Functions & Their Graphs Quiz 2-2: Characteristics of Functions (Part 2) Directions: Determine if the function shown on the graph is continuous. If not, identify the type (infinite, jump, or removable) and location of discontinuity. 1. 2. 3. ❑ Continuous ❑ Continuous ❑ Continuous ❑ Discontinuous ❑ Discontinuous ❑ Discontinuous Type: ____________________ Type: ____________________ Type: ____________________ Location: ________________ Location: _________________ Location: ________________ Directions: Describe the end behavior of each graph. 4. 5. 6. __________________________ ___________________________ __________________________ __________________________ ___________________________ __________________________ Directions: Determine if the relation shown on the graph is symmetrical to the x-axis, y-axis, and/or origin. Verify your answer algebraically in the box. 7. x 2 + y = 7
x-axis


y-axis origin none © Gina Wilson (All Things Algebra®, LLC), 2017 2 2 8. x − y = 4 ❑ x-axis ❑ ❑ ❑ y-axis origin none Directions: Determine algebraically if the function is even, odd, or neither. Show your work in the box. 2 x 4 2 9. f ( x) = 2 3 x 10. f ( x) = 11. f ( x) = −4 x + 13 x − 4 x −1 ❑ ❑ ❑ even ❑ ❑ ❑ odd neither even ❑ ❑ ❑ odd neither even odd neither Directions: Find the average rate of change of each function on the given interval. 3 12. f ( x) = x − 7 x + 3; [−2, 1] 2 13. f ( x) = 2x − 5 ; [−8, − 4] x +1 12. _________________ 13. _________________ 14. _________________ 14. The height of an object thrown straight up from a height 5 feet above the ground is given by the equation h(t) = -16t2 + 42t + 5, where t is the time in seconds after the object is thrown. Find the average velocity of the object from 1.5 to 2.25 seconds. © Gina Wilson (All Things Algebra®, LLC), 2017 PARENT FUNCTIONS Graphic Organizer A function family is a group of functions with similar characteristics. The parent function is the simplest of the functions in a family. You must be able to recognize each of these functions along with their characteristics. LINEAR Parent Function: Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): ABSOLUTE VALUE End Behavior: Parent Function: Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: QUADRATIC Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Parent Function: CUBIC Discontinuities? Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Discontinuities? Even or Odd? © Gina Wilson (All Thing Algebra®, LLC), 2017 SQUARE ROOT Parent Function: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: CUBE ROOT Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: RECIPROCAL Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Parent Function: EXPONENTIAL Domain: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Discontinuities? Even or Odd? © Gina Wilson (All Thing Algebra®, LLC), 2017 GREATEST INTEGER LOGARITHMIC Parent Function: Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: SINE Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: COSINE Parent Function: Discontinuities? Even or Odd? Domain: Range: x-intercept(s): y-intercept: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: Discontinuities? Even or Odd? © Gina Wilson (All Thing Algebra®, LLC), 2017 TRANSFORMATIONS Reference Sheet Transformations affect the appearance of a parent function. There are two main types of transformations: • Rigid Transformations change the position or orientation of a function. Translations and reflections are rigid transformations. • Nonrigid Transformations distort the shape of the function by changing its size. Dilations are nonrigid transformations. The rules for transformations are given below. It is important to memorize these rules! TRANSLATIONS (a shift) REFLECTIONS (a flip) DOWN REFLECTION IN THE X-AXIS: UP REFLECTION IN THE Y-AXIS: LEFT RIGHT DILATIONS (to compress or stretch) VERTICAL DILATIONS HORIZONTAL DILATIONS COMPRESSION COMPRESSION STRETCH STRETCH The function will compress or The function will compress or stretch vertically by a factor of ______! stretch horizontally by a factor of ______! © Gina Wilson (All Thing Algebra®, LLC), 2017 Name: ___________________________________________________ ________________ Date: _________________________________ _________________________________ Topic: Class: _ ___________________________________________________ _________________________________ ________________ _________________________________ Main Ideas/Questions Notes/Examples _ Directions: Given each function, identify both the parent function and the transformations from the parent function. IDENTIFYING TRANSFORMATIONS 1. f ( x)  x3  4 2. f ( x)  2 ( x  1) 2 1  3. f ( x)    x   6 4  5. f ( x)  1 ( x  6)2 2 3 7. f ( x)  4 x  1  2 9. f ( x)   1 ( x  4)  7 3 4. f ( x)  3( x  5)  2 6. f ( x)   1 3 4x 8. f ( x)  x5 1 2 3 10. f ( x)   ( x  3)3  8 2 © Gina W ilson (All Things Algebra®, LLC), 2017 WRITING FUNCTIONS Directions: Using the graph of each function, identify the parent function, then write an equation for the function under each transformation. 11. a) Parent Function: b) Translate 4 units right: c) Reflect in the x-axis and translate 5 units up: 12. a) Parent Function: b) Translate 7 units down and 3 units left: c) Horizontally stretch by a factor of 3 and translate 2 units right: 13. a) Parent Function: b) Translate 1 unit up and vertically stretch by a factor of 4: c) Vertically compress by a factor of 1/3 and translate 6 units left: 14. a) Parent Function: b) Reflect in the y-axis and translate 4 units right: c) Horizontally compress by a factor of 1/2 and translate 5 units down: 15. If the function f ( x) = 2( x − 5)2 + 3 is reflected across the y-axis, then translated right 6 units and down 2 units, write the equation of the new function. 16. A function was vertically stretched by a factor of 2, then translated 2 units left and 5 units up. The resulting function is represented by f ( x) = 6 x + 4 − 1. Write the equation of the original function. © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 6: Parent Functions & Transformations ** This is a 2-page document! ** Directions: Given each function, identify both the parent function and the transformations from the parent function. 2. f (x) = (−x)3 + 7 2 f (x) = x − 4 1. 3 3. f (x) = 6 −1 x 4. f (x) = 4(x − 2) + 7
1  5. f ( x) =  x  + 5  2  6. f ( x) = − 5 7. f ( x) = − ( x + 11)2 2 8. f ( x) = 9. f (x) = 2 −(x − 3) 10. f (x) = 3 −5(x + 9) − 4 x 2 1 +9 x−5 © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Using the graph of each function, identify the parent function, then write an equation for the function under each transformation. a) Parent Function: 11. b) Translate 4 units down and 3 units left: c) Vertically stretch by a factor of 2, then translate 5 units left: 12. a) Parent Function: b) Translate 3 units up and 8 units right: c) Horizontally compress by a factor of ¼, then reflect in the y-axis: 13. a) Parent Function: b) Horizontally stretch by a factor of 3 and translate 4 units down: c) Vertically stretch by a factor of 3/2, reflect in the x-axis, and translate 8 units left: 14. a) Parent Function: b) Vertically compress by a factor of 1/2, then reflect in the y-axis and translate 4 units right: c) Reflect in the x-axis and, then translate 7 units up: 15. If the function f ( x) = 4 x + 1 is horizontally stretched by a factor of 2, reflected across the x-axis, and translated up 3 units, write the equation of the new function. 16. A function was vertically stretched by a factor of 3, then translated 5 units down and 7 units 3 3 left. The resulting function is represented by f ( x) = − ( x + 4) − 2 . Write the equation of the 2 original function. © Gina Wilson (All Things Algebra®, LLC), 2017 GRAPHING FUNCTIONS Directions: Identify the parent function and transformations from the parent function given each function. Then, graph the function using the transformations and identify its key characteristics. 1 f ( x) = − x − 1 + 7 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 2 1  f ( x) =  x  4  Domain: Range: x-intercept(s): y-intercept(s): 2 Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 3 f ( x) = 2 x − 5 Domain: Range: x-intercept(s): y-intercept(s): 3 Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 4 f ( x) = 3 −( x − 2) Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 5 f ( x) = 3 2( x + 1) + 4 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 6 f ( x) = −4 x+3 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 7 f ( x) = 3 − 1 x +4 3 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 8 f ( x) = 1 −1 2x Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 9 f ( x) = 1 x+2 +5 4 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 10 f ( x) = −3( x − 4) + 6 Domain: Range: x-intercept(s): y-intercept(s): 2 Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 11 f ( x) =
x + 1  − 3 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 12 f ( x ) = −2
x − 4  + 1 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 7: Graphing Functions ** This is a 2-page document! ** Directions: Identify the parent function and transformations from the parent function given each function. Then, graph the function and identify its key characteristics. 1. f ( x) = 2( x + 1)3 − 5 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 2. f ( x) = 5 −( x − 6) 2 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 3. f ( x) = − 1 ( x + 3) 3 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 4. f ( x) = − 1 2 ( x − 5) − 3 2 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 5. f ( x ) = 3 3 −2 x + 1 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 6. f ( x) = −1 +3 x−4 Domain: Range: x-intercept(s): y-intercept(s): Parent Function: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples Piecewise Functions Directions: Find each value given the piecewise function. Evaluating Piecewise Functions 2 x−5  2 1. f ( x) = − x + 4 x   2x + 6 a) f (−5) if x < −3 if − 3 ≤ x < 7 if x ≥ 7 b) f (2) c) f (7)  1 3  2 x + 7 if x ≤ −4  2. g ( x) =  8 if − 4 < x < −2 
x − 4  if x ≥ − 2 a) g (−3) Graphing Piecewise b) g (−8) c) g (3.1) Directions: Graph each function, then give the domain, range, and identify the location and type of any discountinuities.  x + 3 − 1 if x < 0 3. f ( x) =   2 x − 5 if x ≥ 0 Functions Domain: Range: Discontinuities: © Gina Wilson (All Things Algebra®, LLC), 2017 −4 − x if x ≤ −3  2  − x if x > −3  2 4. h( x) =  1 Domain: Range: Discontinuities:  4 if x ≤ 1  5. f ( x) =  x + 3 2 x − 7 if x > 1  Domain: Range: Discontinuities: 3 x + 8 if x ≤ −4   6. g ( x) =  − 1 if − 4 < x ≤ 2 ( x − 2) 2 − 5 if x > 2  Domain: Range: Discontinuities:  7 if x < −1  3 7. f ( x) =  x − 1 if − 1 ≤ x ≤ 2   − 2 x + 1 if x > 2 Domain: Range: Discontinuities: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 8: Piecewise Functions ** This is a 2-page document! ** Directions: Find each function value.  x 3  1 if x  2 1. f ( x)   3 x  5 if x  2  x 2  7 if x  7  2. g ( x)   3 x  15 if  7  x  5  6 x if x  5  a) f (4) a) g (7) b) f (2) if  7  x  2  3 x 3. f ( x)    x  4  5 if x  2    4. h( x)      a) f (1.2) a) h(5) b) f (2) b) g (4) 3x  2 if x  5 x  5x 3 x7 2 if  5  x  0 2 if x  0 8 b) h    15  Directions: Graph each function, then give the domain, range, and identify the location and type of any discountinuities.  1 2 if x  3  x 5. f ( x)   2  10  x if x  3 Domain: Range: Discontinuities: © Gina Wilson (All Things Algebra®, LLC), 2017  2  x  1 6. h( x)    2 x4  3 if x  3 if x  3 Domain: Range: Discontinuities:  3 x  4 if x  2   2 if  2  x  2 7. g ( x)    x 2  11 if x  2  Domain: Range: Discontinuities: ì -(x + 3) if x < -4 ï 8. h(x) = í 2x - 7 if - 4 < x < 1 ï if x ³ 1 î3x -1 Domain: Range: Discontinuities:  x  3 9. f ( x)  ( x  3)  5  2 x 1  if x  5 if  5  x  2 if x  2 Domain: Range: Discontinuities: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: _________________________________________________ Pre-Calculus Date: ______________________________Per: ________ Unit 2: Functions & Their Graphs Quiz 2-3: Parent Functions, Transformations, Graphing 1. Which parent functions are odd and continuous? Check all that apply. ❑ Cubic ❑ Reciprocal ❑ Greatest Integer ❑ Absolute Value ❑ Square Root ❑ Exponential ❑ Sine ❑ Quadratic ❑ Cube Root ❑ Logarithmic ❑ Cosine ❑ Linear 2. Use the graph below to answer each part. a) Give the equation for the parent function. b) Suppose this function is horizontally compressed by a factor of 1/3, reflected over the x-axis, then translated 5 units up. Write an equation to represent the new function. 3. Describe all the transformations from the parent function: f ( x)  ( x  2) 4 3 ______________________________________________________________________________________________________ ______________________________________________________________________________________________________ Identify the parent function and transformations given each function. Then, graph the function and identify the key characteristics. 4. f ( x)  1 x 1 7 2 Parent Function: Transformations: Domain: Range: x-int(s): y-int: Extrema: Increasing Interval(s): Decreasing Interval(s): End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 3 5. f ( x)  4 x  2 Parent Function: Domain: Range: x-int(s): y-int: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: 6. f ( x)  3 1 x5 Parent Function: Domain: Range: x-int(s): y-int: Extrema: Transformations: Increasing Interval(s): Decreasing Interval(s): End Behavior: Use the following piecewise function for questions 7-8: 7. Evaluate for each value: a) f(-7) = ( x  5)3  4 if x  3  f ( x)    x2  2 if  3  x  2   1 x if x  2  b) f(-3) = 8. Graph, then give the domain, range, and identify the location and type of any discontinuities. Domain: Range: Discontinuities: © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________________________ ________________ Date: _________________________________ _________________________________ Topic: Class: _ ___________________________________________________ _________________________________ ________________ _________________________________ Main Ideas/Questions Notes/Examples _ Functions can be combined using addition, subtraction, multiplication, and division to create a new function. Operations with Functions The Domain of Combined Functions Example 1: f(x) = x2 + x; g(x) = 3x – 1 Example 2: Notation Addition ( f  g )( x) Subtraction ( f  g )( x) Multiplication ( f  g )( x) Division  f    ( x) g Equivalent Operation The domain of the new function is the intersection of the combined functions. The domain is further restricted with the quotient function to exclude values in which g ( x)  0. Given f(x) and g(x), find each new function and state its domain. a) ( f  g )( x) b) ( f  g )( x) c) ( f  g )( x)  f  d)   ( x) g a) ( f  g )( x) b) ( f  g )( x) c) ( f  g )( x)  f  d)   ( x) g f ( x)  x2  9 ; g( x)  x © Gina Wilson (All Things Algebra®, LLC), 2017 Example 3: a) ( g  f )( x) b) ( f  g )( x) c) ( f  g )( x)  f  d)   ( x) g a) ( f  g )( x) b) ( g  f )( x) c) ( g  f )( x)  f  d)   ( x) g a) ( g  f )( x) b) ( f  g )( x) c) ( g  f )( x)  f  d)   ( x) g f ( x)  2 x3 ; g ( x)  x2  4 x Example 4: 2 1 f ( x)  ; g ( x)  x x2 Example 5: f ( x)  x  6 ; g ( x)  x2 Example 6: f ( x)  4  x2 ; g ( x)  2 x  8 Given f(x) and g(x), evaluate each function. a) ( f  g )(5) b) ( f  g )( 2) © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions COMPOSITION of FUNCTIONS Notes/Examples Another method to combine functions is called a composition. Given f(x) and g(x), the composite function ( f
g )( x ) is defined as: The domain of the composite function ( f
g )( x) is the domain of g such that g(x) is in the domain of f. THE DOMAIN OF Domain of g ( f
g )( x) EXAMPLE 1: 2 f ( x) = x + 7 ; g ( x) = x − 4 EXAMPLE 2: f ( x) = 3 − x ; g ( x) = x Domain of f g g(x) f f(g(x)) For examples 1-5: Given f(x) and g(x), find each function and state its domain. a) ( f
g )( x) b) ( g
f )( x) a) ( f
g )( x) b) ( g
f )( x) a) ( f
g )( x) b) ( g
f )( x) x x +1 EXAMPLE 3: 3 f ( x) = 9 − 2 x ; g ( x) = 4 x + 1 © Gina Wilson (All Things Algebra®, LLC), 2017 EXAMPLE 4: f ( x) = x + 5 ; g ( x) = x 3 3 f ( x) = − 2 x ; g ( x) = x + 1 b) ( g
f )( x) For examples 6-7: Given f(x) and g(x), evaluate each function. a) ( f
g )(3) b) ( g
f )(−5) a) ( f
g )(9) EXAMPLE 7: 2 a) ( f
g )( x) x+4 EXAMPLE 6: f ( x) = x + 5 x ; g ( x) = b) ( g
f )( x) 2 EXAMPLE 5: f ( x) = x − 1; g ( x) = a) ( f
g )( x) b) ( g
f )(7) x−4 Directions: Find two functions f and g such that ( f
g )( x ) = h( x ) . DECOMPOSING A FUNCTION 8. h( x) = (1− 2 x) 10. h( x) = 3 4 −1 x+7 9. h( x) = 3 2 x +1 11. h( x) = − 1 x+5 +3 2 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 9: Function Operations & Compositions of Functions ** This is a 2-page document! ** Using the given functions, find each function and state its domain. 1. f ( x) = x 2 + 2 x and g ( x) = 5 − x a) ( f + g )( x) b) ( g − f )( x) c) ( f ⋅ g )( x) g d)   ( x)  f  2 4 x3 and g ( x) = x 5 a) ( g + f )( x) 2. f ( x) = − c) ( f ⋅ g )( x) 3. f (x) = x − 7 and g ( x) = 2 x a) ( f + g )( x) c) ( g ⋅ f )( x) b) ( g − f )( x)  f  d)   ( x) g b) (g − f )(x) g d)   ( x)  f  © Gina Wilson (All Things Algebra®, LLC), 2017 x −1 x b) (h
g )( x) 4. f ( x) = x 2 − 5 x , g ( x) = 3 x + 1, and h( x) = a) ( g
f )( x) c) ( f
g )( x) 5. f ( x) = x 3 + 6 , g ( x) = 3 4 x , and h( x) = 2 x − 5 a) (h
f )( x) b) (h
g)(x) c) (g
f )(x) Use the given functions to evalaute each function. 6. f ( x) = x 2 − 4 x , g ( x) = 2 − 3 x and h( x) = 3 x a) ( f − g )(−5) b) ( g ⋅ f )(3) c) ( g
h)( −6) d) (h
f )(2) Find two functions f and g such that ( f
g)( x) = h( x) . 7. h( x) = 2(3 x − 1)3 8. h( x) = − 3 +7 x+5 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions Notes/Examples • What is the inverse of a relation? _______________________________________ Inverse Relations _________________________________________________________________________ • Notation: The inverse of a relation R is written as ________________ • Example: Give the inverse of relation R below. R: {(0, -7), (-1, 2), (-4, -2), (6, 3)} R-1: __________________________________ One-to-One Functions • Recall: The vertical line test is used to determine whether a relation is a function given a graph. Horizontal Line Test • The horizontal line test is used to determine whether the _____________ of a relation is a function. • BOTH the relation and its inverse must be a function in order to be a one-to-one function. Directions: Determine whether the relations are one-to-one functions. Examples 1. 2. 3. 4. 5. 6. © Gina Wilson (All Things Algebra®, LLC), 2017 • Notation: The inverse of a function f(x) is written as ________________ Inverse Functions • Not all functions have inverse functions. • To find the inverse of a function: Replace f(x) with y and y • Examples h Interchange x h Solve for y h Rewrite y as f-1(x). Directions: Determine if f(x) has an inverse, if yes, find f-1(x). State any restrictions in the domain. 7. f ( x) = 4 x − 8 1 3 8. f ( x) = x − 1 2 2 5 x−8 9. f ( x) = ( x − 4) + 6 10. f ( x) = 11. f ( x) = − x + 2 − 7 12. f ( x) = − 3 13. f ( x) = x−9 4 3 x +1 2 2 14. f ( x) = 3 x − 1; x ≥ 0 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: Date: ___________________________________________________ _________________________________ Topic: Class: ___________________________________________________ _________________________________ Main Ideas/Questions VERIFYING INVERSES Graphically Notes/Examples Directions: Find the inverse of each function. Verify graphically. 1. f ( x) = 3 x − 6 3 2. f ( x) = x + 4 VERIFYING INVERSES Algebraically Directions: Verify algebraically that f and g are inverse functions. 3. f ( x) = x−7 and g ( x) = 2 x + 7 2 2 4. f ( x) = x − 5 (if x ≥ 0) and g ( x) = x+5 © Gina Wilson (All Things Algebra®, LLC), 2017 5. f ( x) = More Examples 6. f ( x) = 3 x +2 3 x+3 and g ( x) = 2x − 1 2x Directions: Find the inverse of each function. State any restrictions in the domain. Verify graphically and algebraically. Verify Graphically Verify Algebraically 2 7. f ( x ) = ( x + 4) − 2 (if x ≥ -4) 8. f ( x) = 5 +2 x 9. f (x) = 1 x+3 2 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: ___________________________________ Unit 2: Functions & Their Graphs Date: ________________________ Per: ______ Homework 10: Inverse Relations & Functions, Verifying Inverses ** This is a 2-page document! ** Directions: Determine whether the relations are one-to-one functions. 1. 2. 3. Directions: Determine if f(x) has an inverse, if yes, find f-1(x). State any restrictions in the domain. 4. f (x) = 3 x+3 4 5. f (x) = (x − 5)2 + 9 6. f (x) = 1 x+2 7. f (x) = 8. f (x) = 4 2x −1 9. f (x) = 3 x + 7 −1 2x + 5 x −7 © Gina Wilson (All Things Algebra®, LLC), 2017 Directions: Find the inverse of each function. State any restrictions in the domain. Verify graphically and algebraically. 10. f (x) = 11. f (x) = −2(x − 1) 12. f (x) = 13. f (x) = Verify Graphically x−4 3 Verify Algebraically 3 3 x+4 x +2 −7 © Gina Wilson (All Things Algebra®, LLC), 2017 Name: _________________________________________________ Pre-Calculus Date: ______________________________Per: ________ Unit 2: Functions & Their Graphs Quiz 2-4: Function Operations, Compositions, and Inverses Given f(x) = x2 + 3x – 10, g(x) = 6 – x3 , and h(x) = 2x + 1, find each function and state its domain. 1. ( f  h)( x) 2. (h  g )( x) 1. _____________________________ D: __________________________ 2. _____________________________ 2 3. ( f  g )( x) h 4.   ( x) f  D: __________________________ 3. _____________________________ D: __________________________ 5. ( f h)( x) 4. _____________________________ 6. (h g )( x) D: __________________________ 5. _____________________________ 7. Given f ( x)  x  3 and g ( x)  state its domain. x  6 , find ( f  g )( x) and D: __________________________ 6. _____________________________ D: __________________________ 1 x 1 8. Given f ( x)  and g ( x)  , find ( f  g )( x) and x 2 state its domain. 7. _____________________________ D: __________________________ 8. _____________________________ D: __________________________ 9. Given f ( x)  x 2  4 and g ( x)  x  1 , find ( f g )( x) and state its domain. 9. _____________________________ D: __________________________ 10. ____________________________ 10. Given h( x)  ( x  5)3  2 , find two functions, f and g, such that ( f g )( x)  h( x). ____________________________ © Gina Wilson (All Things Algebra®, LLC), 2017 Given f ( x )  2  5 x , g ( x )  x2 3 , and h( x )   x  7 , evaluate each function. x2 12. ( g  h)(8) 11. ( f  g )(6) 11. _______________ 12. _______________ g 13.   (4) f  14. ( f 13. _______________ h)(1) 14. _______________ Given f(x), find f-1(x). State any restrictions in the domain. x 6 2 15. f ( x)  ( x  1)  3 ; x  1 16. f ( x)  x2 15. ___________________________ 16. ___________________________ Verify graphically that f(x) and g(x) are inverse functions. 17. f ( x)  2 x  5 and g ( x)  x5 2 18. f ( x)  x  4 and g ( x)  ( x  4)2 ; x  4 Determine algebraically using compositions whether functions f and g are inverse functions. 3 19. f ( x)  2 x  6 and g ( x)  3 x6 2 Inverses? 20. f ( x)   yes  no 4 4  2 and g ( x)  x x2 Inverses?  yes  no © Gina Wilson (All Things Algebra®, LLC), 2017 Unit 2 Test Study Guide Name: __________________________________________ (Functions & Their Graphs) Date: ____________________________ Per: __________ Topic 1: Evaluating Functions For questions 1 and 2, evaluate given f ( x )  1. f (9) x2 . 2x  3 2. f ( x  1) For questions 3 and 4, evaluate given g ( x )  3 x  x 2 . 3. g (2 x  1) 4. g (3 x)  4 x  7 if x  3 For questions 5 and 6, evaluate given h( x )   . 3 2   x  2 x if x  3 5. h  7  6. h(3) Topic 2: Parent Functions, Transformations, and Graphing For each function family below, give the parent function and sketch the shape of its graph. 7. Linear 8. Absolute Value 9. Quadratic 10. Cubic 11. Square Root 12. Cube Root 13. Reciprocal 14. Greatest Integer © Gina W ilson (All Things Algebra®, LLC), 2017 15. If the quadratic parent function is reflected in the y-axis and vertically compressed by a factor of ½, write an equation to represent the new function. 16. If the cube root parent function is horizontally stretched by a factor of 4, then translated 5 units right and 3 units up, write an equation to represent the new function. 17. The absolute value parent function has transformations applied such that it creates an absolute maximum at (-2, 7). Write an equation that could represent this new function. 18. A certain function is vertically stretched by a factor of 2, horizontally compressed by ¼, and translated down 6. The new function is represented by f ( x )  6
8 x   2 . Write the equation of the original function. 19. Describe all transformations from the parent function given the function below. 20. Describe all transformations from the parent function given the function below. 3 1  f ( x)  3  x   7 2  f ( x)  2 ( x  5)  2 3 Graph each function and identify all key characteristics. Domain: 3 1 21. f ( x)  x4 Range: x-int: y-int: Extrema Increasing Interval: Decreasing Interval: End Behavior: 22. f ( x)  2( x  3)2  4 Domain: Range: x-int: y-int: Extrema Increasing Interval: Decreasing Interval: End Behavior: © Gina Wilson (All Things Algebra®, LLC), 2017 23. f ( x)  2 3 ( x  4)  3 Domain: Range: x-int: y-int: Extrema Increasing Interval: Decreasing Interval: End Behavior: Topic 3: Piecewise Functions Identify the domain and range of each graph below. State the location and type of any discontinuties. Domain: Domain: 24. 25. Range: Range: Discontinuities: Discontinuities: 26. Graph the function below. Identify the domain and range, then, state the location and type of any discontinuties.  3  2 x  6  f ( x)    x   x 1 3  Domain: Range: if x  4 if  4  x  1 if x  1 Discontinuities: Topic 4: Average Rate of Change Find the average rate of change of the function on the given interval. 2x  1 27. f ( x)  2 x 2  3 x  1; [-3, 2] ; [-10, -5] 28. f ( x)  x3 © Gina W ilson (All Things Algebra®, LLC), 2017 29. A football is kicked from a point on the ground such that its height h(t ), in feet, is given by the equation h(t )  16 t 2  80t , where t is time in seconds. Find the average rate of change in the height of the ball from when it reaches its maximum height until it reaches the ground. Topic 5: Tests for Symmetry / Even & Odd Functions Use the graph to determine if the relations given below are symmetrical to the x-axis, y-axis, and/or origin. Confirm your answer algebraically. 31. y  2 x  5 30. x 2  y 2  4 Determine whether the function below is even, odd, or neither. Prove your answer algebraically. 32. f ( x)  3 x 3  5 x 33. f ( x)  5 x 2  2 x  1 Topic 6: Function Operations & Compositions of Functions Use f ( x )  3  2 x , g( x )  x  7 , and h( x )  x 2  5 x to find each function below. Be sure to state any domain restrictions, wherever necessary. 34.  g  f  ( x) 35.  h  f  ( x) f  36.   ( x) h © Gina W ilson (All Things Algebra®, LLC), 2017 Use f ( x )   x 2  2 x , g( x )  domain for each. 37. (h f )( x) x  7 , and h( x)  3 x  1 to find each function below. Give the 38. ( f 39. ( f g )( x) Given h(x) below, find two functions, f and g, such that ( f 40. h( x)  5 2 x9 g )( x )  h( x ). 41. h( x)   2( x  5)  7 Use f ( x)  10  2 x , g( x )  3 2 x  3 , and h( x)  42. ( g  f )(15) h)( x) h 43.   (12) g 1 x  5 to evaluate each function below. 2 44. ( g h)  6  Topic 7: Inverse Functions Determine if the graph represents a one-to-one function. 45. 46. 47. © Gina W ilson (All Things Algebra®, LLC), 2017 Determine if f(x) has an inverse, if yes, find f-1(x). State any restrictions in the domain. 49. f ( x)  2 x  5 48. f ( x)  3 x  7  2 50. f ( x)  4 x 2  7; x  0 51. f ( x)  x 6 x5 Prove f (x) and g(x) are inverses both algebraically and graphically. 4 52. f ( x)   2 x 4 g ( x)  x2 3 1  53. f ( x)   x   3 2  g ( x)  2 3 x  3 © Gina W ilson (All Things Algebra®, LLC), 2017 Na