Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematics of radio engineering wikipedia , lookup
Continuous function wikipedia , lookup
Dirac delta function wikipedia , lookup
Non-standard calculus wikipedia , lookup
History of the function concept wikipedia , lookup
Function (mathematics) wikipedia , lookup
Elementary mathematics wikipedia , lookup
Signal-flow graph wikipedia , lookup
The Reciprocal Function Family 8-2 Vocabulary Review 1. Place a ✓ in the box if the fractions are reciprocals. Place an ✗ in the box if they are NOT reciprocals. ✓ 7 8 8 and 7 ✗ 1 1 x and 2x ✓ 1 3x and 3x 2. The reciprocal of a positive number is positive. Yes / No The reciprocal of a negative number is positive. Yes / No A negative number has no reciprocal. Yes / No The reciprocal of 0 is undefined. Yes / No reciprocal function (noun) rih SIP ruh kul FUNGK shun Related Terms: reciprocal (noun), multiplicative inverse (noun), asymptote (noun), improper fraction (noun) Main Idea: A reciprocal function is any function that contains a fraction with the independent variable in the denominator. 5 1 1 Examples: y 5 3x 1 1, f (x) 5 x 2 2 , g(x) 5 2x 2 8 x Nonexamples: y 5 x 2 2, f (x) 5 2 , g(x) 5 x21 4x 5 , h(x) 5 8x 1 5 Use Your Vocabulary Complete each equation with the given expressions to make a reciprocal function. 3. 5 and 3x y5 6 3x Chapter 8 4. 7 and x 5. (x 1 1) and 2 2 7 1 5 6. (x 2 8) and 1 f (x) 5 y5 x11 x 214 1 23 f (x) 5 x28 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Vocabulary Builder Problem 1 Graphing an Inverse Variation Function Got It? What is the graph of y 5 12 x ? Identify the x- and y-intercepts and the asymptotes of the graph. Also, state the domain and range of the function. 7. Circle the graph that shows y 5 12 x. y y y 8 8 8 4 4 4 x 8 4 O 4 x 8 8 4 O 4 x 8 8 4 O 4 4 4 8 8 8 4 8 8. Underline the correct word or words to complete the sentences about the graph. The value of x cannot be 0, so there is no x- / y- intercept. The value of y cannot be 0, so there is no x- / y- intercept. The x-axis is the horizontal / vertical asymptote. The y-axis is the horizontal / vertical asymptote. 9. The domain is the set of all real numbers except x 5 0 . Key Concepts The Reciprocal Function Family Parent function: y 5 1x , x 2 0 stretches by a factor of a. shrinks by a factor of a. reflects across the x-axis. shifts horizontally h units and has a shifts vertically k units x h 215 Combined y k; x h. vertical asymptote at x h. horizontal asymptote at y k. a x h k, where x h 2 2 The graph of y 1 x 2 Translations occur based on the values of h and k in y 1 2 a x 2 a 1, When 0 a 1, the graph of y a 0, a x , x 0. 2 2 Stretches and compressions occur based on the value of a in y 2 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. The range is the set of all real numbers except y 5 0 . Lesson 8-2 Problem 3 Graphing a Translation 1 Got It? What is the graph of y 5 x 2 4 1 6? Identify the domain and range. 10. Find the equations of the asymptotes. Then sketch the asymptotes on the graph. a 1 y xh k x4 6 The vertical asymptote is x 4 . The horizontal asymptote is y 6 . 11. Graph the parent function, y 5 x1 , on the coordinate plane at the right. y 8 6 4 2 12. Translate the points (1, 1) and (21, 21) from the graph of y 5 1x . Each point moves 4 units to the right and 6 units up. 1 13. Draw the branches of y 5 x 2 4 1 6 through the translated points. 14. Find the domain and range. Domain: the set of all real numbers except x 5 Range: the set of all real numbers except y 5 6 Writing the Equation of a Transformation Got It? This graph of a function is a translation of the graph of y 5 2x . 4 What is an equation for the function? 15. Identify the asymptotes: x 5 So h 5 x 1 and y 5 24 8 4 y5x2h1k y5 2 8 1 24 x2 1 17. The simplified form of the equation is y 5 2 x21 24 . Problem 5 Using a Reciprocal Function Got It? The junior class is renting a laser tag facility with a capacity of 325 people. The cost for the facility is $1200. The party must have 13 adult chaperones. If every student who attends shares the facility cost equally, what function models the cost per student C with respect to the number of students n who attend? What is the domain of the function? How many students must attend to make the cost per student no more than $7.50? Chapter 8 O 1 and k 5 24 . 16. Substitute the values of h and k into the general form of a translated reciprocal function. a x 2 4 6 8 216 4 8 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. Problem 4 4 8 6 4 2 O 2 4 6 8 18. Complete the model. C, cost per is student Relate n, the number of students who attend $1200 divided by Capacity Number of chaperones 1200 Write C 1200 or n C n 19. The domain of this function is #n# 1 20. Label the functions y 5 7.5 and y 5 312 . 1200 x in the graph at the right. y 7.5 21. For the cost per student to be no more than $7.50, 160 Intersection X160 Y7.5 students must attend. y 1200 x Lesson Check • Do you UNDERSTAND? Write an equation of the reflection of the graph y 5 1x across the x-axis. 22. Assume y 5 ax is a reflection across the x-axis of y 5 1x . Circle the true statement. Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved. a must be a fraction between 0 and 1. a must be negative. 23. Complete the equation with values of your choosing so the graph is a reflection of y 5 x1 across the x-axis. Place a zero in any box you choose to leave blank. y5 21 1 x2 The value of the numerator must be negative. Sample values are given. 0 0 Math Success Check off the vocabulary words that you understand. reciprocal function asymptote translation Rate how well you can graph translations of reciprocal functions. Need to review 0 2 4 6 8 Now I get it! 10 217 Lesson 8-2