Model Inverse Variation. p Write and graph inverse variation equations. VOCABULARY. Inverse variation. Constant of variation. Branches of a hyperbola

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1 12.1 Model Inverse Variation Goal p Write and graph inverse variation equations. Your Notes VOCABULARY Inverse variation Constant of variation Hperbola Branches of a hperbola Asmptotes of a hperbola Eample 1 Tell whether the equation represents direct variation, inverse variation, or neither. a Write original equation. 5 Divide each side b. Because 5 22 be written in the form 5 } a, 5 22 represents. The constant of variation is. b. } Identif direct and inverse variation 5 Write original equation. 5 Multipl each side b. Because } 5 be written in the form 5 a, } 5 represents. Copright Holt McDougal. All rights reserved. Lesson 12.1 Algebra 1 Notetaking Guide 0

2 12.1 Model Inverse Variation Goal p Write and graph inverse variation equations. Your Notes VOCABULARY Inverse variation The variables and show inverse variation if 5 } a and a Þ 0. Constant of variation The nonzero number a in the equation 5 a } Hperbola The graph of the inverse variation equation 5 } a (a Þ 0) is a hperbola. Branches of a hperbola The two smmetrical parts of a hperbola Asmptotes of a hperbola A line that the hperbola approaches but does not intersect Eample 1 Tell whether the equation represents direct variation, inverse variation, or neither. a Write original equation } Divide each side b. Because 5 22 can be written in the form 5 a }, 5 22 represents inverse variation. The constant of variation is 22. b. } 5 Write original equation. 5 Multipl each side b. Because } 5 can be written in the form 5 a, 5 represents direct variation. } Identif direct and inverse variation Copright Holt McDougal. All rights reserved. Lesson 12.1 Algebra 1 Notetaking Guide 0

3 Checkpoint Tell whether the equation represents direct variation, inverse variation, or neither. 1. } Eample 2 Graph an inverse variation equation Graph 5 } 22. Then find the domain and range for the function. Step 1 Make a table b choosing several integer values of and finding the values of. Then plot the points. To see how the function behaves for values of ver close to 0 and ver far from 0, make a second table for such values and plot the points. Note that is undefined when 5 0. There is no point (0, ) on the graph. Also, there is no value of for which 5 0, so there is no point (, 0) on the graph. Neither the domain nor the range of an inverse variation function includes Step 2 Connect the points in Quadrant II b drawing a smooth curve through them. Repeat for points in Quadrant IV. Both the domain and range for the function are all real numbers ecept 0. 0 Lesson 12.1 Algebra 1 Notetaking Guide Copright Holt McDougal. All rights reserved.

4 Checkpoint Tell whether the equation represents direct variation, inverse variation, or neither. 1. } direct neither inverse variation variation Eample 2 Graph an inverse variation equation Graph 5 } 22. Then find the domain and range for the function. Step 1 Make a table b choosing several integer values of and finding the values of. Then plot the points. To see how the function behaves for values of ver close to 0 and ver far from 0, make a second table for such values and plot the points. Note that is undefined when 5 0. There is no point (0, ) on the graph. Also, there is no value of for which 5 0, so there is no point (, 0) on the graph. Neither the domain nor the range of an inverse variation function includes undef Step 2 Connect the points in Quadrant II b drawing a smooth curve through them. Repeat for points in Quadrant IV. Both the domain and range for the function are all real numbers ecept 0. 0 Lesson 12.1 Algebra 1 Notetaking Guide Copright Holt McDougal. All rights reserved.

5 GRAPHS OF DIRECT VARIATION AND INVERSE VARIATION EQUATIONS Direct Variation 5 a, a > 0 5 a, a < 0 Inverse Variation 5 a }, a > 0 5 a }, a < 0 Eample Use an inverse variation equation The variables and var inversel, and 5 2 when 5 6. Write an inverse variation equation that relates and. Find the value of when 5. Solution Because varies with, the equation has the form 5 } a. Use the fact that 5 6 and 5 2 to find the value of a. 5 a } 5 a Write inverse variation equation. Substitute for and for. 5 a Multipl each side b. An equation that relates and is 5. When 5, 5 5. Copright Holt McDougal. All rights reserved. Lesson 12.1 Algebra 1 Notetaking Guide 05

6 GRAPHS OF DIRECT VARIATION AND INVERSE VARIATION EQUATIONS Direct Variation 5 a, a > 0 5 a, a < 0 Inverse Variation 5 a }, a > 0 5 a }, a < 0 Eample The variables and var inversel, and 5 2 when 5 6. Write an inverse variation equation that relates and. Find the value of when 5. Solution Because varies inversel with, the equation has the form 5 } a. Use the fact that 5 6 and 5 2 to find the value of a. 5 a } 2 5 a 6 Use an inverse variation equation Write inverse variation equation. Substitute 6 for and 2 for a Multipl each side b 6. An equation that relates and is 5 22 }. When 5, Copright Holt McDougal. All rights reserved. Lesson 12.1 Algebra 1 Notetaking Guide 05

7 Eample Write an inverse variation equation Tell whether the ordered pairs (25, 1.2), (22, ), (1.5, 2), (8, 20.75), (10, 20.6) represent inverse variation. If so, write the inverse variation equation. Solution Find the products for all pairs (, ): 25(1.2) 5, 22() 5, 1.5(2) 5, 8(20.75) 5, 10(20.6) 5 The products are equal to the same number,. So,. The inverse variation equation is 5, or 5. Checkpoint Complete the following eercises.. Graph 5 }. Then find the domain and range The variables and var inversel, and 5 5 when 5 2. Write an inverse variation equation that relates and. Then find the value of when 5 9. Homework 6. Tell whether the ordered pairs (220, 2), (212, 25), (10, 6), (15, ), (0, 1.5) represent inverse variation. If so, write the inverse variation equation. 06 Lesson 12.1 Algebra 1 Notetaking Guide Copright Holt McDougal. All rights reserved.

8 Eample Write an inverse variation equation Tell whether the ordered pairs (25, 1.2), (22, ), (1.5, 2), (8, 20.75), (10, 20.6) represent inverse variation. If so, write the inverse variation equation. Solution Find the products for all pairs (, ): 25(1.2) 5 26, 22() 5 26, 1.5(2) 5 26, 8(20.75) 5 26, 10(20.6) 5 26 The products are equal to the same number, 26. So, varies inversel with. The inverse variation equation is 5 26, or 5 } 26. Checkpoint Complete the following eercises.. Graph 5 }. Then find the domain and range. Both the domain and range are all real numbers ecept zero The variables and var inversel, and 5 5 when 5 2. Write an inverse variation equation that relates and. Then find the value of when } 215 ; 5 2 } 5 Homework 6. Tell whether the ordered pairs (220, 2), (212, 25), (10, 6), (15, ), (0, 1.5) represent inverse variation. If so, write the inverse variation equation. varies inversel with ; 5 60 } 06 Lesson 12.1 Algebra 1 Notetaking Guide Copright Holt McDougal. All rights reserved.