An exponential function is non linear function of the form
y = ab^{x}, where a ≠ 0 and b ≠ 1 and b > 0
Horizontal asymptote :
Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.
For the function in the form
y = ab^{x}
horizontal asymptote is y = 0
For the function in the form
y = ab^{x-h} + k
horizontal asymptote is y = k
Domain :
For exponential function, domain is all set of real values.
Range :
Set of possible outputs is range. If there is no reflection
y = (0, ∞)
If it reflection across y-axis, then domain will be
y = (k, -∞)
Find the following characteristics of the exponential functions given below.
1) Graph the function 2) a = ? 3) b = ? 4) y - intercept 5) x - intercept |
6) Domain 7) Range 8) Asymptote 9) Increasing interval 10) Decreasing interval |
Problem 1 :
y = 2^{x}
Solution :
a = 1, b = 2 > 0 (exponential growth)
Horizontal asymptote is y = 0.
Put x = 0, y = 2^{0}
So, y-intercept is (0, 1).
1) Graph the function
2) a = 1 3) b = 2 4) y - intercept = 1 5) x - intercept = none 6) Domain = (-∞, ∞) 7) Range = (0, ∞) |
8) Asymptote, y = 0 9) Increasing interval (-∞, ∞) 10) Decreasing interval None |
Problem 2 :
y = 4^{x} - 1
a) Describe the transformations to the parent function.
b) What is the asymptote?
c) What is the domain and range?
d) What is the interval of increase/decrease?
e) What are the intercepts?
f) What is the end behavior?
Solution :
y = 4^{x} - 1
Comparing the given function with parent function y = ab^{x}
a) The graph of y = 4^{x} should be moved one unit down to get the graph of y = 4^{x} - 1.
b) Horizontal asymptote is y = -1
c) Domain is all real values. Range is (-1, ∞)
d) Since it is an exponential growth function (b = 4 > 1), increasing interval is (-∞, ∞).
e)
x-intercept : Put y = 0 0 = 4^{x} - 1 4^{x} = 1 There is no x-intercept. |
y-intercept : put x = 0 y = 4^{0} - 1 y = 1 - 1 y = 0 |
g) What is the end behavior :
When x --> -∞, then y --> -1
When x --> ∞, then y --> ∞
Problem 3 :
y = 2(1/2)^{x}
a) Describe the transformations to the parent function.
b) What is the asymptote?
c) What is the domain and range?
d) What is the interval of increase/decrease?
e) What are the intercepts?
f) What is the end behavior?
Solution :
Here a = 2 and b = 1/2 (0 < b < 1)
So, it is exponential decay function.
Since a > 1, vertical stretch will be there.
b) Horizontal asymptote is at y = 0
c) Domain is (-∞, ∞) and range (0, ∞).
d) b = 1/2 <1, it is exponential decay and its decreasing interval is at (-∞, ∞).
e) There is no x-intercept and when y = 0, x = 2.
f)
When x --> -∞, then y --> ∞
When x --> ∞, then y --> 0
Problem 4 :
y = 2^{-x }+ 2
a) Describe the transformations to the parent function.
b) What is the asymptote?
c) What is the domain and range?
d) What is the interval of increase/decrease?
e) What are the intercepts?
Solution :
Here a = 1 and b = 2 (b > 1)
So, it is exponential growth function.
a) Comparing with the parent function y = 2^{x}, there is reflection across y-axis and move the graph 2 units up.
b) Horizontal asymptote at y = 2
c) Domain is all real values (-∞, ∞) and range (2, ∞).
d) There us no x-intercept and y-intercept is 2.
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