Answer:
Option D. The rate of change of function 2 is less than the rate of change of the function 1
Step-by-step explanation:
step 1
Find the rate of change and the y-intercept of the function 1
we have
[tex]y=30x+19[/tex]
This is the equation of the line in slope intercept form
where
the slope is m=30
the y-intercept is b=19
Remember that the slope is the same that the rate of change
so
The rate of change of the function 1 is [tex]m_1=30[/tex]
The y-intercept of the function 1 is [tex]b_1=19[/tex]
step 2
Find the slope of the function 2
From the data in the table take two points
(-12,-311) and (1,14)
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
substitute the values
[tex]m=\frac{14+311}{1+12}[/tex]
[tex]m=\frac{325}{13}[/tex]
[tex]m=25[/tex]
step 3
Find the equation of the function 2 in slope intercept form
[tex]y=mx+b[/tex]
we have
[tex]m=25[/tex]
[tex]point\ (1,14)[/tex]
substitute
[tex]14=25(1)+b[/tex]
solve for b
[tex]b=14-25[/tex]
[tex]b=-11[/tex]
The equation of the function 2 is
[tex]y=25x-11[/tex]
so
The rate of change of the function 2 is [tex]m_2=25[/tex]
The y-intercept of the function 2 is [tex]b_2=-11[/tex]
Verify each statement
case A) Function 2 can be described by the equation y=25x-100
The statement is false
Because the equation of the function 2 is [tex]y=25x-11[/tex]
case B) Function 2 can be described by the equation y=13x-11
The statement is false
Because the equation of the function 2 is [tex]y=25x-11[/tex]
case C) The y-intercept of function 1 is less than the y-intercept of the function 2
The statement is false
Because, the y-intercept of function 1 is greater than the y-intercept of the function 2
[tex]19 > -11[/tex]
case D) The rate of change of function 2 is less than the rate of change of the function 1
The statement is true
[tex]25 <30[/tex]
