Reteach Variation Functions

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1 8-1 Variation Functions The variable y varies directly as the variable if y k for some constant k. To solve direct variation problems: k is called the constant of variation. Use the known and y values in the problem to solve for k. Write the direct variation equation, substituting the value for k. Use the direct variation equation to solve for the missing variable. If y varies directly as, and y 52 when 4, find y when 6. Step 1 Use y 52 when 4. y k 52 k 4 13 k Step 2 Write the direct variation equation. y k y 13 Step 3 Solve for y when 6. y 13 y 13 6 y 78 The variable y varies jointly as the variables and z if y kz for some constant k. Joint variation problems are solved like direct variation problems. If y varies jointly as and z, and y 90 when 36 and z 5, find y when 40 and z 3. Step 1 y kz 90 k k 0.5 k Step 2 Write the joint variation equation. y kz y 0.5z Step 3 Solve for y when 40 and z 3. y 0.5z y y 60 Solve each problem. 1. If y varies directly as, and y 30 when 20, find y when 50. a. Step 1: b. Step 2: c. Step 3: y k y 1.5 y k 20 y k 1.5 y If y varies jointly as and z, and y 150 when 2.5 and z 12, find y when 4 and z 6.5. a. Step 1: b. Step 2: c. Step 3: y kz ; 150 k ; k, k 5 y 5z y 5z ; y ; y Holt Algebra 2

2 8-1 Variation Functions (continued) The variable y varies inversely as the variable if y k for some constant k. If y varies inversely as, and y 4 when 30, find y when 20. Step 1 Use y 4 when 30. Step 2 Write the inverse variation equation. Step 3 Solve for y when 20. y k 4 k y k y k y 120 y y 6 To graph the inverse variation function y 120, make a table of values. y y Because the function is undefined for 0, make separate tables for negative and positive -values. Solve each problem. 3. If y varies inversely as, and y 2 when 9, find y when 6. Then graph the inverse variation function. a. Step 1: y k ; 2 k ; k 18 9 b. Step 2: y 18 c. Step 3: y 18 ; y 18 6 ; y 3 7 Holt Algebra 2

3 Solving Rational Equations and Inequalities To solve a rational equation, clear any denominators by multiplying each term on both sides of the equation by the least common denominator, LCD. Solve: Step 1 The LCD is. Multiply each term by This makes the equation a quadratic equation. Set one side equal to 0 to solve a quadratic equation. Step 5 Set each factor equal to Step Check Z Z Always check the solutions to rational equations , 2 3, 2 5, 1 38 Holt Algebra 2

4 Solving Rational Equations and Inequalities (continued) Check all solutions to rational equations. If the solution to a rational equation makes the denominator equal to zero, then that solution is NOT a solution. It is called an etraneous solution. Solve: Step 1 The LCD is 2 6. Multiply each term by Remember to multiply EVERY term by the LCD Step 5 Set each factor equal to 0 and solve Step 6 Check: ? Z The only solution is 2. 6 is etraneous. This value makes the denominators of the original equation equal to ; ; 3 39 Holt Algebra 2

5 Solving Rational Equations and Inequalities To solve a rational equation, clear any denominators by multiplying each term on both sides of the equation by the least common denominator, LCD. Solve: Step 1 The LCD is. Multiply each term by This makes the equation a quadratic equation. Set one side equal to 0 to solve a quadratic equation. Step 5 Set each factor equal to Step Check Z Z Always check the solutions to rational equations , 2 3, 2 5, 1 38 Holt Algebra 2

6 Solving Rational Equations and Inequalities (continued) Check all solutions to rational equations. If the solution to a rational equation makes the denominator equal to zero, then that solution is NOT a solution. It is called an etraneous solution. Solve: Step 1 The LCD is 2 6. Multiply each term by Remember to multiply EVERY term by the LCD Step 5 Set each factor equal to 0 and solve Step 6 Check: ? Z The only solution is 2. 6 is etraneous. This value makes the denominators of the original equation equal to ; ; 3 39 Holt Algebra 2

7 Reading Strategy Analyze Information The solutions to a rational function are those values that result from solving the equation. Some solutions, however, are etraneous. Etraneous solutions are solutions of a derived equation that are not solutions of the original equation. Find the solution. Write the equation Substitute the solution into the original equation Analyze. The solution gives a denominator of 0, so the solution is etraneous. This equation has no solution. Identify values of that would be etraneous solutions for each equation , 0, 1 5, , , 1 4 2, 1 Solve. 10. Ralph solved the inequality He found the solutions to be 1 and 1 2. He knows the solution has to be epressed as an inequality. He thinks the solution should be written 1 or 1. Is he correct? How do you know? 2 No; possible answer: the solution should be 1 or is an etraneous solution Holt Algebra 2