2.5 Linear vs. Exponential

Linear functions A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear Function. A linear function is a first degree polynomial of the form, y = m x + b, where m and b are constants and x is a real variable. The constant m is called slope and b is called y-intercept.

Exponential Functions The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e. percentage increase or decrease) in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases. The graph always lies above the x-axis graph

Example 1 A population has size 1500 at time t=0, with t in years. a) If the population decreases by 75 people per year, find a formula for the population, P, at time t. b) If the population decreases by 5% per year, find a formula for the population, P, at time t.

Example 2 The following formulas give the populations (in thousands) of four different cities, A, B, C, and D. Which are changing exponentially? Describe in words how each of these populations is changing over time (in years).

Example 3 Go to wkst 2.5 for example problem

Example 4 Let P(t) be the population of a city, in thousands, t years after 2002, with P(4)=300 and P(9)=475. ◦Find a linear formula for P(t). Describe in words the city’s annual population growth. ◦Find an exponential formula for P(t). Describe in words the city’s annual population growth.