Answer:
[tex]\begin{array}{ccccccccc}{Hours} & {0} & {1} & {2} & {3} & {4}& {5} & {6} & {7} \ \\ {Persons} & {1} & {2} & {4} & {8} & {16}& {32} & {64} & {128} \ \end{array}[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Solving (a): Complete the table
Let
[tex]x = hours[/tex]
[tex]y = persons[/tex]
From the table question, we have:
[tex](x_1,y_1) = (0,1)[/tex]
[tex](x_2,y_2) = (1,2)[/tex]
[tex](x_3,y_3) = (2,4)[/tex]
The pattern follows that, an increment in x by doubles the value of 1.
So, the other values are:
[tex](x_4,y_4) = (3,8)[/tex]
[tex](x_5,y_5) = (4,16)[/tex]
[tex](x_6,y_6) = (5,32)[/tex]
[tex](x_7,y_7) = (6,64)[/tex]
[tex](x_8,y_8) = (7,128)[/tex]
So, the complete table is:
[tex]\begin{array}{ccccccccc}{Hours} & {0} & {1} & {2} & {3} & {4}& {5} & {6} & {7} \ \\ {Persons} & {1} & {2} & {4} & {8} & {16}& {32} & {64} & {128} \ \end{array}[/tex]
Solving (b): The graph
The table follows an exponential function:
[tex]y = ab^x[/tex]
We have: [tex](x_1,y_1) = (0,1)[/tex]
This gives:
[tex]y = ab^x[/tex]
[tex]1 = ab^0[/tex]
[tex]b^0 \to 1[/tex]
So:
[tex]1 = a*1[/tex]
[tex]1 = a[/tex]
[tex]a =1[/tex]
Also: [tex](x_5,y_5) = (4,16)[/tex]
This gives:
[tex]y = ab^x[/tex]
[tex]16 = 1 * b^4[/tex]
[tex]16 = b^4[/tex]
[tex]16 \to 2^4[/tex]
So:
[tex]2^4 = b^4[/tex]
Cancel the exponents (4)
[tex]2 =b[/tex]
[tex]b = 2[/tex]
So, the function [tex]y = ab^x[/tex] is:
[tex]y = 2^x[/tex]
See attachment 2 for graph
