a. The given table is
Notice, the value of x increases at equal intervals of 1
Also, the value of y increases at an equal interval of 3
This means for the y values the difference between consecutive terms is 3
Also, for the x values, the difference between consecutive terms is 1
Hence, the table represents a linear function
The general form of a linear function is
[tex]y=mx+c[/tex]
Where m is the slope
From the interval increase
[tex]m=\frac{\Delta y}{\Delta x}=\frac{3}{1}=3[/tex]
Hence, m = 3
The equation becomes
[tex]y=3x+c[/tex]
To get c, consider the values
x = 0 and y = 2
Thi implies
[tex]\begin{gathered} 2=3(0)+c \\ c=2 \end{gathered}[/tex]
Hence, the equation of the linear function is
[tex]y=3x+2[/tex]
b. The given table is
Following the same procedure as in (a), it can be seen that there is no constant increase in the values of y
Hence, the function is not linear
This implies that the function is exponential
The general form of an exponential function is given as
[tex]y=a\cdot b^x[/tex]
Consider the values
x =0, y = 3
Substitute x = 0, y = 3 into the equation
This gives
[tex]\begin{gathered} 3=a\times b^0 \\ \Rightarrow a=3 \end{gathered}[/tex]
The equation become
[tex]y=3\cdot b^x[/tex]
Consider the values
x =1, y = 6
Substitute x = 1, y = 6 into the equation
This gives
[tex]\begin{gathered} 6=3\cdot b^1 \\ \Rightarrow b=\frac{6}{3}=2 \end{gathered}[/tex]
Therefore the equation of the exponential function is
[tex]y=3\cdot2^x[/tex]
c. The given table is
As with (b) above,
The function is exponential
Using
[tex]y=a\cdot b^x[/tex]
When
x = 0, y = 10
This implies
[tex]\begin{gathered} 10=a\cdot b^0 \\ \Rightarrow a=10 \end{gathered}[/tex]
The equation becomes
[tex]y=10\cdot b^x[/tex]
Also, when
x = 1, y =5
The equation becomes
[tex]\begin{gathered} 5=10\cdot b^1 \\ \Rightarrow b=\frac{5}{10} \\ b=\frac{1}{2} \end{gathered}[/tex]
Therefore, the equation of the exponential function is
[tex]y=10\cdot(\frac{1}{2})^x[/tex]
