What is horizontal asymptote ?
The horizontal line which is very closer to the curve is known as horizontal asymptote.
Exponential function will be in the form
y = ab^{x - h }+ k
Equation of horizontal asymptote will be y = k
From the graph, to find equation of horizontal asymptote we have to follow the steps below.
Find the horizontal asymptote of the following exponential functions.
Problem 1 :
y = 2(4/7)^{x} - 4
Solution :
Comparing the given equation with y = ab^{x - h }+ k
y = k is the equation of horizontal asymptote.
Then, the required horizontal asymptote y = -4
Problem 2 :
Find the equation for the horizontal asymptote of the following function
Solution :
While observing the graph from bottom to top, the line drawn at the point y = 5 is closer to the curve. So, equation of horizontal asymptote is y = 5.
Problem 3 :
identify the domain and range for each graphed exponential function.
Solution :
To fix domain of the function graphed above, we have to look at the possible outcomes.
So, the domain is all real values.
To fix range of the function, we have to find horizontal asymptote. Here x-axis or y = 0 is closer to the curve.
So, the range is (-∞, 0), the equation of horizontal asymptote is y = 0.
Problem 4 :
identify the domain and range for each graphed exponential function.
Solution :
To fix domain of the function graphed above, we have to look at the possible outcomes.
So, the domain is all real values.
To fix range of the function, we have to find horizontal asymptote. Here x-axis or y = -4 is closer to the curve.
So, the range is (-4, ∞), the equation of horizontal asymptote is y = -4.
Problem 5 :
identify the domain and range for each graphed exponential function.
Solution :
To fix domain of the function graphed above, we have to look at the possible outcomes.
So, the domain is all real values.
To fix range of the function, we have to find horizontal asymptote. Here x-axis or y = 1 is closer to the curve.
So, the range is (1, ∞), the equation of horizontal asymptote is y = 1.
Problem 6 :
Find the equation of the horizontal asymptote from the table given below.
Solution :
To find the equation of horizontal asymptote, we have to find the equation of exponential function.
Choosing two points from the table,
(0, 4) and (1, 8)
y = ab^{x}
y = ab^{x} Applying the point (0, 4) 4 = ab^{0} 4 = a(1) a = 4 |
Applying the point (1, 8) 8 = ab^{1} ab = 8 Applying a = 4 b = 8/4 b = 2 |
By applying the value of a and b, we get
y = 4(2)^{x}
Equation of horizontal asymptote is y = 0.
Problem 7 :
Solution :
To find the equation of horizontal asymptote, we have to find the equation of exponential function.
Choosing two points from the table,
(0, 1) and (1, 5)
y = ab^{x}
y = ab^{x} Applying the point (0, 1) 1 = ab^{0} 1 = a(1) a = 1 |
Applying the point (1, 5) 5 = ab^{1} ab = 5 Applying a = 1 b = 5 |
By applying the value of a and b, we get
y = 1(5)^{x}
Equation of horizontal asymptote is y = 0.
For each exponential function.
Problem 8 :
y = 2^{x} + 3
Solution :
x = -2 y = 2^{-2}+3 y = 1/4 + 3 y = 13/4 |
x = 0 y = 2^{0}+3 y = 1 + 3 y = 4 |
x = 2 y = 2^{2}+3 y = 4+3 y = 7 |
x = 4 y = 2^{4}+3 y = 16+3 y = 19 |
x -2 0 2 4 |
y 13/4 4 7 19 |
Problem 9 :
y = 2(3)^{x} - 5
Solution :
x = -2 y = 2^{-3}-5 y = (1/8)-5 y = -39/8 |
x = 0 y = 2^{0}-5 y = 1-5 y = -4 |
x = 2 y = 2^{2}-5 y = 4-5 y = -1 |
x = 4 y = 2^{-4}-5 y = (1/16)-5 y = -79/16 |
Problem 10 :
Write the exponential equation in the form y = ab^{x }and find horizontal asymptote.
Solution :
Choosing two points from the graph,
(0, 3) and (-1, 18)
y = ab^{x}
(0, 3) 3 = ab^{0} 3 = a(1) a = 3 |
(-1, 18) 18 = ab^{-1} 18 = a/b b = 18/3 b = 6 |
y = 3(6)^{x}
Equation of horizontal asymptote is y = 0.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM