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§7.6 Exponential Functions. Warm-up. Graph each function. 1. y = 3 x 2 . y = 4 x 3 . y = –2 x Simplify each expression. 3 2 5 . 5 –3 6 . 2 • 3 4 7. 2 • 3 –2 8 . 3 • 2 –1 9 . 10 • 3 2. 1 5 • 5 • 5. 1 5 3. 1 125. Solutions. 1. y = 3 x. 3. y = –2 x.
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Warm-up Graph each function. 1.y = 3x2.y = 4x3.y = –2x Simplify each expression. 325. 5–36. 2 • 34 7. 2 • 3–28. 3 • 2–19. 10 • 32
1 5 • 5 • 5 1 53 1 125 Solutions 1.y = 3x 3.y = –2x 2.y = 4x 4. 32 = 3 • 3 = 9 5. 5–3 = = = 6. 2 • 34 = 2 • (3 • 3 • 3 • 3) = 2 • 81 = 162 7. 2 • 3–2 = 2 • = 2 • = 8. 3 • 2-1 = 3 • = 3 • = or 1.5 9. 10 • 32 = 10 • (3 • 3) = 10 • 9 = 90 1 9 2 9 1 32 1 21 3 2 1 2
Key Terms Exponential Function: Exponential Function:an exponential function is a function in the form y = a •bx, where a is a nonzero constant, b is greater that 0 and not equal to 1, and x is a real number. Exponential Function:an exponential function is a function in the form y = a •bx, where a is a nonzero constant, b is greater than 0 and not equal to 1, and x is a real number. Examples:y = 0.2 • 3x f(x) = 4 • 0.6x
1 16 3 16 3 16 –2 3 • 4–2 = 3 • = Examples 1 & 2: Identifying and Evaluating an Exponential Function Evaluate each exponential function. a.y = 3x for x = 2, 3, 4 c.p(q) = 3 • 4q for the domain {–2, 3} xy = 3xy qp(q) = 3 • 4qp(q) 2 32 = 9 9 3 33 = 27 27 4 34 = 81 81 3 3 • 43 = 3 • 64 = 192 192 b.y = 3x for x = 2, 3, 4 xy = 3xy 2 3(2) = 66 3 3(3) = 99 4 3(4) = 1212
In two years, there are 8 three-month time periods. ƒ(x) = 2 • 48 ƒ(x) = 2 • 65,536 Simplify powers. ƒ(x) = 131,072 Simplify. Example ¥: Real-World Problem Solving Suppose two mice live in a barn. If the number of mice quadruples every 3 months, how many mice will be in the barn after 2 years? ƒ(x) = 2 • 4x
2 9 2 3 2 2 9 –2 2 • 3–2 = = (–2, ) 2 31 2 3 2 3 –1 2 • 3–1 = = (–1, ) 0 2 • 30 = 2 • 1 = 2 (0, 2) 1 2 • 31 = 2 • 3 = 6 (1, 6) 2 2 • 32 = 2 • 9 = 18 (2, 18) Example 3: Graphing an Exponential Function Graph y = 2 • 3x. xy = 2 • 3x (x, y)
1 1.251= 1.25 1.3 (1, 1.3) 2 1.252 = 1.5625 1.6 (2, 1.6) 3 1.253= 1.9531 2.0 (3, 2.0) 4 1.254= 2.4414 2.4 (4, 2.4) 5 1.255 = 3.0518 3.1 (5, 3.1) Example 4: Graphing an Exponential Model The function ƒ(x) = 1.25x models the increase in size of an image being copied over and over at 125% on a photocopier. Graph the function. x ƒ(x) = 1.25x (x, ƒ(x))
