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Question

write an exponential equation for each of the following cases.

a) An increasing exponential function with a y intercept of 2.

b) A decreasing exponential function with a y intercept of 5.

c) An exponential function with a y intercept of 9 and passes through point (1,3).

d) An exponential function that passes through the point (1,10) and (2,50).

Answer 1

c)

Let basic exponential function is y = ab^x    --------------(1)

Y-intercept is 9. So it (y = ab^x) passes through point (0,9).

9 = ab⁰  ⇒   a = 9

Equation (1) become y = 9b^x     --------------(2)

y = 9b^x passes through point (1,3).

3 = 9b¹   ⇒   b = 3/9  ⇒  b = 1/3

Equation (2) become y = 9(1/3)^x

Solution :  y = 9(1/3)^x.

Answer 2

a)

Let basic increasing exponential function is y = ab^x  ( b>1)

Y-intercept is 2. So it (y = ab^x) passes through point (0,2).

2 = ab⁰  ⇒  a = 2

y = 2b^x

An increasing exponential function with a y intercept of 2 is y = 2b^x  where b>1.

Answer 3

b)

Let basic decreasing exponential function is y = ab^-x   ( b>1)

Y-intercept is 5. So it (y = ab^-x) passes through point (0,5).

5 = ab^-0 ⇒ a = 5

y = 5b^-x

An decreasing exponential function with a y intercept of 5 is y = 5b^-x where b>1.

Answer 4

d)

Let basic exponential function is y = ab^x        --------------(1)

y = ab^x is passing through point (1,10).

10 = ab¹   ⇒   ab = 10                                 --------------(2)

y = ab^x  is also passing through point (2,50).

50 = ab²   ⇒   ab² = 50                                --------------(3)

 

(3)/(2)   ⇒   ab²/ab = 50/10  ⇒   b = 5.

Equation(2)  ⇒   a(5) = 10 ⇒  a = 10/5 ⇒  a = 2.

Substitute a=2,b=5 in Equation(1).

y = 2(5)^x

Solution :  y = 2(5)^x