Answer:
Part 1) The rate of change is [tex]\frac{17}{15}[/tex]
Part 2) The initial value is 68
Part 3) The function rule to the linear model is [tex]y=\frac{17}{15}x+68[/tex]
Step-by-step explanation:
we know that
The linear equation in slope intercept form is equal to
[tex]y=mx+b[/tex]
where
m is the slope or unit rate
b is the y-intercept or initial value
step 1
Find the slope
take two points from the table
(0,68) and (15,85)
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{85-68}{15-0}[/tex]
[tex]m=\frac{17}{15}[/tex]
In a linear function , the slope is the same that the rate of change
therefore
The rate of change is [tex]\frac{17}{15}[/tex]
step 2
Find the y-intercept
we know that
The y-intercept is the value of y when the value of x is equal to zero
Looking at the table
For x=0, y=68
therefore
The y-intercept is
[tex]b=68[/tex]
The y-intercept is also called the initial value
therefore
The initial value is 68
step 3
Determine the function rule to the linear model
[tex]y=mx+b[/tex]
we have
[tex]m=\frac{17}{15}[/tex]
[tex]b=68[/tex]
substitute
[tex]y=\frac{17}{15}x+68[/tex]