Answer :
Answer:
option B and option C represent an exponential function.
Step-by-step explanation:
We know that following are some of the properties of different functions.
Linear Function:
The constant change in x and y
Quadratic Function:
The constant change in x
The constant 2nd differences in y
Exponential Function:
The constant change in x
The constant ratio in y
Determining the nature of the exponential function
Option A)
x 0 1 2 3 4
y 0 1 4 9 16
Here:
There is a constant change in x:
1 - 0 = 1; 2 - 1 = 1; 3 - 2 = 1; 4 - 3 = 1
BUT, there is NO CONSTANT ratio, as
1/0 = ∞, 4/1 = 4, 9/4 ; 16/9
Thus,
Table A DOES NOT represents an exponential function.
Option B)
x 0 1 2 3 4
y 1/3 1 3 9 27
Here:
There is a constant change in x:
1 - 0 = 1; 2 - 1 = 1; 3 - 2 = 1; 4 - 3 = 1
There is also a constant ratio, as
1/[1/3] = 3, 3/1 = 3, 9/3 = 3; 27/9 = 3
As there is constant change in x as well a constant ratio.
Thus,
Table B REPRESENTS an exponential function.
Option C)
x 0 1 2 3 4
y 5 5/2 5/4 5/8 5/16
Here:
There is a constant change in x:
1 - 0 = 1; 2 - 1 = 1; 3 - 2 = 1; 4 - 3 = 1
There is also a constant ratio, as
[5/2]/5= 1/2,
[5/4]/[5/2] = 1/2
[5/8]/[5/4] = 1/2
[5/16]/[5/8] = 1/2
As there is constant change in x as well a constant ratio.
Thus,
Table C REPRESENTS an exponential function.
Option D)
x 0 1 2 3 4
y 4.5 4 3.5 3 2.5
Here:
There is a constant change in x:
1 - 0 = 1; 2 - 1 = 1; 3 - 2 = 1; 4 - 3 = 1
BUT, there is NO CONSTANT ratio, as
4/4.5 = 4/[45/10] = 40/45 = 8/9
3.5/4 = [35/10]/4 = 35/40 = 7/8
3/3.5 = 3/[35/10] = 30/35 = 6/7
2.5/3 = [25/10]/3 = 25/30 = 5/6
Table D DOES NOT represent an exponential function.
Conclusion:
Therefore, option B and option C represent an exponential function.
The functions that are exponential would be:
B). [tex]x[/tex] 0 1 2 3 4
[tex]y[/tex] 1/3 1 3 9 27
C). [tex]x[/tex] 0 1 2 3 4
[tex]y[/tex] 5 5/2 5/4 5/8 5/16
What are Exponential Functions?
B). Given that,
[tex]x[/tex] 0 1 2 3 4
[tex]y[/tex] 1/3 1 3 9 27
As we see,
The value of [tex]x[/tex] undergoes a constant change of 1
[tex]1 - 0 = 1; \\2 - 1 = 1; \\3 - 2 = 1; \\4 - 3 = 1[/tex]
The ratio also remains constant
[tex]1/[1/3] = 3, \\3/1 = 3, \\9/3 = 3; \\27/9 = 3[/tex]
Thus, it exemplifies an exponential function.
Similarly,
C). [tex]x[/tex] 0 1 2 3 4
[tex]y[/tex] 5 5/2 5/4 5/8 5/16
As we see,
The value of [tex]x[/tex] undergoes a constant change of 1
[tex]1 - 0 = 1; \\2 - 1 = 1; \\3 - 2 = 1; \\4 - 3 = 1[/tex]
The ratio also remains constant
[tex][5/2]/5= 1/2\\\\[5/4]/[5/2] = 1/2\\\\[5/8]/[5/4] = 1/2\\\\[5/16]/[5/8] = 1/2[/tex]
Thus, it exemplifies an exponential function.
Thus, options B and C are the correct answers as the other two have a constant change but not a constant ratio.
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