Solving f(x) = g(x) Algebraically Unit 4 Lesson 5 Notes 2 Solving f(x) = g(x) Algebraically
Example: Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount? If x is the number of years and y is the weight, write an equation that represents the bunny’s weight (y) after a given number of years (x). Bunny: y = 3x + 5
Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year Barbara has a bunny that weighs 5 pounds and gains 3 pounds per year. Her cat weighs 19 pounds and gains 1 pound per year. When will the bunny and the cat weigh the same amount? If x is the number of years and y is the weight, write an equation that represents the cat’s weight (y) after a given number of years (x). Cat: y = 1x + 19
The Bunny and Cat’s weight will be equal after 7 years. When two equations are equal to the same thing (or variable) then you can put them equal to each other and solve. -1x -x 2x + 5 = 19 - 5 - 5 2x = 14 x = 7 Bunny: y = 3x + 5 Cat: y = x + 19 3x + 5 = x + 19 The Bunny and Cat’s weight will be equal after 7 years.
Solving f(x) = g(x) Algebraically Replace any function notation f(x), g(x), with y = Place the two functions equal to each other. Looking at what type of function you have, solve the equation using appropriate method.
Solving Different types of Equations Quadratic: If there is an x2 and NO x term, get x2 by itself and take square root of both sides. If there is an x2 AND an x term, get one side equal to zero, factor , put equal to zero and then solve for x. Absolute Value: Get |x| by itself on one side of equal sign. Make two equations and set equal to positive answer AND negative answer. Solve for x. Linear: Get x to one side of equal sign. If a constant is with x, add/subtract that value to both sides. If a number is being multiplied by x, divide both sides by that number.
Example 1: Find the solution to f(x) = g(x). f(x) = 7x – 3 and g(x) = x + 15 (same as y = 7x – 3 and y = x + 15) 7x – 3 = x + 17 6x – 3 = 15 6x = 18 x = 3 Both are linear functions
Example 2: Find the solution to f(x) = g(x). f(x) = x2 – 4 and g(x) = 45 x2 – 4 = 45 x2 = 49 x = 7 and -7 Quadratic but NO ‘x’ term
Example 3: Find the solution to f(x) = g(x). f(x) = x2 – 5x and g(x) = 2x – 10 x2 – 5x = 2x – 10 x2 – 7x + 10 = 0 (x – 5)(x – 2) = 0 x = 5 and x = 2 Quadratic - with x2 AND ‘x’ term
Example 4: Find the solution to f(x) = g(x). f(x) = |x – 19| and g(x) = 22 |x – 19| = 22 x – 19 = 22 x – 19 = -22 x = 41 and x = -3 Absolute Value Function