Answer:
Part 1) The rate of change is 
Part 2) The initial value is 68
Part 3) The function rule to the linear model is 
Step-by-step explanation:
we know that
The linear equation in slope intercept form is equal to
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
substitute the values
In a linear function , the slope is the same that the rate of change
therefore
The rate of change is
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
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
we have
substitute
(32-n)470
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58057, 58058, 58059 are the numbers!
1. m
2. One set of ordered pairs
3. b
To show why this is, I’m going to explain how to write the equation for a linear function with two random sets of ordered pairs - (1,0) and (5, 8).
First, find the slope. The formula for slope is m = (y2 - y1)/(x2-x1) where m is the slope and (x1, y1) and (x2, y2) are two sets of points.
This is why #1 is m. M is the letter used when finding slope.
To find m, I plug in the two sets of ordered pairs.
m = (8-0)/(5-1)
m = 8/4
m = 2
An equation for a line (linear function) is written in something called slope-intercept form. It looks like y = mx + b. m is the slope and b is the y-intercept (number y equals when x = 0). If m = 3 and b = 1, the equation would be y = 3x + 1.
Here, you have just solved for m and know it equals 2. Plug that value in for m.
y = 2x + b
To solve for b, plug one ordered pair in for x and y. I will use (1,0)
0 = 2(1) + b
0 = 2 + b
-2 = b
Now that you know b = -2, plug that in for b.
y = 2x - 2. Now you have the equation fo the line.
