Write and solve a real world problem in which you need to find the initial value of the function. then explain how you solved the problem

you are planting roses

you started at 12 noon

at 3pm, you have 80 roses left

at 6pm, you have 20 roses left

if you planted roses at a constant rate, how many roses did you start with?

so we can find the slope of this linear function as (20roses-80roses)/(6pm-3pm)=60roses/3hrs=-20roses/hr

your quantity goes down by 20 roses per hour (you plant 20 per hour)

r=-20t+?

find ?, the number of roses when t=0

at 3pm yo have 80 roses

80=-20(3)+?

80=-60+?

140=?

you had 140 roses intiailly

if you wanted an exponenntial

the town polpulationn grows at an exponential rate

the town was founded in 2000

in 2010, the town's population was 1000

in 2015, the town's populationw as 2000

find intial polpulation

exponential can be modeled by where y=population, t=time and b and a are constants (a=initial population)

we are given that after 10 years (2010), y=102400

after 15 years, (2015), y=3276800

so

and

divide the 2nd by the first equation, remembering our exponential properties like

take 5th root of both sides

subsitute back

the intial population is 100 people

4x^2 - 8x - 13 = 0...divide everything by 4
x^2 - 2x - 13/4 = 0...add 13/4 to both sides
x^2 - 2x = 13/4 ...take half of ur x term, square it, and add it to both sides
x^2 - 2x + 1 = 13/4 + 1
(x - 1)^2 = 17/4
x - 1 = (+-) sqrt (17/4)
x = 1 (+-) sqrt (17/4)

x = 1 + (sqrt 17) / 2 <==
x = 1 - (sqrt 17) / 2 <==

I dont know what steps she made, but here is how I would do it.