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Identify the domain, range, intercept, and asymptote of each exponential function. Then describe the end...

Identify the domain, range, intercept, and asymptote of each exponential function. Then describe the end behavior. SEE EXAMPLE 1.
14. f(x) = 5 * 3^x
15. f(x) = 0.75 * (3/2)^x
16. f(x) = 4 * (1/2)^x
17. f(x) = 7 * 2^(-x)
Identify the transformation from the parent function f(x) = 0.25^x. Then graph the function. SEE EXAMPLE 2
18. g(x) = -6(0.25)^x + 4
19. h(x) = 0.7(0.25)^(-x) - 5
20. Write a function g(x) that represents the exponential function f(x) = 2^x after a vertical stretch of 6 and a reflection across the x-axis. Graph both functions. SEE EXAMPLE 2
Determine whether each function represents exponential growth or decay. Write the base in terms of the rate of growth or decay; identify r, and interpret the rate of growth or decay. SEE EXAMPLES 3 AND 4
21. y = 1000 * 2.5^x
22. f(x) = 10,200 * (3/5)^x
23. f(x) = 12,000 * (7/10)^x
24. y = 450 * 2^x

Answer

  1. 14. f(x) = 5 * 3^x

    Domain: All real numbers
    Range: y > 0
    Intercept: (0, 5)
    Asymptote: y = 0
    End behavior: As x → ∞, f(x) → ∞; as x → -∞, f(x) → 0.

  2. 15. f(x) = 0.75 * (3/2)^x

    Domain: All real numbers
    Range: y > 0
    Intercept: (0, 0.75)
    Asymptote: y = 0
    End behavior: As x → ∞, f(x) → ∞; as x → -∞, f(x) → 0.

  3. 16. f(x) = 4 * (1/2)^x

    Domain: All real numbers
    Range: y > 0
    Intercept: (0, 4)
    Asymptote: y = 0
    End behavior: As x → ∞, f(x) → 0; as x → -∞, f(x) → ∞.

  4. 17. f(x) = 7 * 2^(-x)

    Domain: All real numbers
    Range: y > 0
    Intercept: (0, 7)
    Asymptote: y = 0
    End behavior: As x → ∞, f(x) → 0; as x → -∞, f(x) → ∞.

  5. 18. g(x) = -6(0.25)^x + 4

    Transformation: Vertical stretch by 6, reflection across the x-axis, vertical shift up 4.

  6. 19. h(x) = 0.7(0.25)^(-x) - 5

    Transformation: Reflection across the y-axis, vertical stretch by 0.7, vertical shift down 5.

  7. 20.

    g(x) = -6 * 2^x
    Transformation: Vertical stretch by 6, reflection across the x-axis.

  8. 21. y = 1000 * 2.5^x

    Exponential Growth
    Base: 2.5
    Rate of growth (r): 1.5 (150%)

  9. 22. f(x) = 10,200 * (3/5)^x

    Exponential Decay
    Base: 0.6
    Rate of decay (r): 0.4 (40%)

  10. 23. f(x) = 12,000 * (7/10)^x

    Exponential Decay
    Base: 0.7
    Rate of decay (r): 0.3 (30%)

  11. 24. y = 450 * 2^x

    Exponential Growth
    Base: 2
    Rate of growth (r): 1 (100%)

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