Linear function increase by a constant slope but exponential equations increase by a constant exponent or power.
Linear function will be in the form
y = mx + b
Exponential function will be in the form
y = ab^{x}
(or)
y = ab^{x-h} + k
Linear function |
Exponential function | |
Describe each function as a linear function or exponential function.
Problem 1 :
Solution :
Values of x is increasing by 1, values of y
1 x 2 = 2
2 x 2 = 4
4 x 2 = 8
Every values of y is multiplied by 2. Since the multiplication factor is the same, it must be exponential function.
Finding the exponential function :
y = ab^{x }----(1)
b = 2
From the table, applying the point (0, 4), we get
4 = a(2)^{-0}
4 = a(1)
a = 4
By applying the value of a and b in (1), we get
y = 4(2)^{x }
Problem 2 :
Solution :
Values of x is increasing by 1, values of y
(1/25) / (1/5) = 1/5
(1/5) / 1 = 1/5
1/5
Every values of y is multiplied by 1/5. Since the multiplication factor is the same, it must be exponential function.
Finding the exponential function :
y = ab^{x }----(1)
b = 1/5
From the table, applying the point (0, 1), we get
1 = a(1/5)^{-0}
1 = a(1)
a = 1
By applying the value of a and b in (1), we get
y = 1(1/5)^{x }
y = (1/5)^{x }
Problem 3 :
y = -6x + 9
Solution :
y = -6x + 9
The given function is in the form,
y = mx + b
Here m = -6 and b = 9
Problem 4 :
Solution :
Considering the values of x, it is increased by 1.
Considering the values of y, it is increased by 3.
Finding the linear function :
y = mx + b
m = 3
y = 3x + b ----(1)
Applying the point (0, 22), we get
22 = 3(0) + b
22 = 0 + b
b = 22
Applying the value of b in (1), we get
y = 3x + 22
Problem 5 :
y = 3 (2)^{x }
Solution :
The given function in the form, y = ab^{x}. Then it is exponential function.
Problem 6 :
Solution :
The given graph is the graph of exponential function.
y = ab^{x - h }+ k
Here y = k is the horizontal asymptote. From the given graph, horizontal asymptote is y = 4
k = 4
By applying the of k, we get
y = ab^{x - h }+ 4
By choosing three points from the graph, we can get the values of a, b and h.
Problem 7 :
Select which exponential function below models this situation: Lupe bought a gold chain for $350, and the value of the chain increases by 4% per year. Use 𝑦 for the value of the chain and 𝑡 for the time in years.
A) 𝑦 = 350^{4𝑥} B) 𝑦 = 350(4)^{𝑥}
C) 𝑦 = 350(1.4)^{𝑥} D) 𝑦 = 350(1.04)^{𝑥}
Solution :
Every year the value of the chain increase by the constant factor.
Since the growth factor is the same, it must be a exponential function.
y = a(1 + r%)^{x}
Growth factor = 4%
a = 350
y = 350(1 + 4%)^{x}
= 350(1 + 0.04)^{x}
= 350(1.04)^{x}
So, option D is correct.
Problem 8 :
Solution :
The values of x is increasing by 1. By observing the values of y, it is increased by 7.
Constant rate of change is 7,
Creating exponential function :
y = mx + b
y = 7x + b --------(1)
By choosing the point (0, 3) and applying this point in the above function
3 = 7(0) + b
3 = 0 + b
b = 3
By applying the value of b in (1), we get
y = 7x + 3
Problem 9 :
Solution :
By observing the values of x is increased by 1, by observing the values of y, we get
32/16 = 2
16/8 = 2
8/4 = 2
Since the constant factor is the same, it must be an exponential function.
y = ab^{x }----(1)
Applying the point (0, 32), we get
32 = ab^{0 }
32 = a(1)
a = 32
Applying the value of a in (1), we get
y = 32b^{x}
Applying the point (1, 16), we get
16 = 32(b)^{1}
16/32 = b
b = 1/2
y = 32(1/2)^{x}
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM