# 6.8 Fitting exponential models to data

 Page 1 / 12
In this section, you will:
• Build an exponential model from data.
• Build a logarithmic model from data.
• Build a logistic model from data.

In previous sections of this chapter, we were either given a function explicitly to graph or evaluate, or we were given a set of points that were guaranteed to lie on the curve. Then we used algebra to find the equation that fit the points exactly. In this section, we use a modeling technique called regression analysis to find a curve that models data collected from real-world observations. With regression analysis , we don’t expect all the points to lie perfectly on the curve. The idea is to find a model that best fits the data. Then we use the model to make predictions about future events.

Do not be confused by the word model . In mathematics, we often use the terms function , equation , and model interchangeably, even though they each have their own formal definition. The term model is typically used to indicate that the equation or function approximates a real-world situation.

We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review. Take a moment to rethink each of these functions, reflect on the work we’ve done so far, and then explore the ways regression is used to model real-world phenomena.

## Building an exponential model from data

As we’ve learned, there are a multitude of situations that can be modeled by exponential functions, such as investment growth, radioactive decay, atmospheric pressure changes, and temperatures of a cooling object. What do these phenomena have in common? For one thing, all the models either increase or decrease as time moves forward. But that’s not the whole story. It’s the way data increase or decrease that helps us determine whether it is best modeled by an exponential equation. Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay.

Recall that exponential functions have the form $\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.\text{\hspace{0.17em}}$ When performing regression analysis, we use the form most commonly used on graphing utilities, $\text{\hspace{0.17em}}y=a{b}^{x}.\text{\hspace{0.17em}}$ Take a moment to reflect on the characteristics we’ve already learned about the exponential function $\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ (assume $\text{\hspace{0.17em}}a>0\right):$

• $b\text{\hspace{0.17em}}$ must be greater than zero and not equal to one.
• The initial value of the model is $\text{\hspace{0.17em}}y=a.$
• If $\text{\hspace{0.17em}}b>1,$ the function models exponential growth. As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases, the outputs of the model increase slowly at first, but then increase more and more rapidly, without bound.
• If $\text{\hspace{0.17em}}0 the function models exponential decay . As $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases, the outputs for the model decrease rapidly at first and then level off to become asymptotic to the x -axis. In other words, the outputs never become equal to or less than zero.

the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
if a=3, b =4 and c=5 find the six trigonometric value sin
pls how do I factorize x⁴+x³-7x²-x+6=0
in a function the input value is called
how do I test for values on the number line
if a=4 b=4 then a+b=
a+b+2ab
Kin
commulative principle
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
x^20+x^15+x^10+x^5/x^2+1
evaluate each algebraic expression. 2x+×_2 if ×=5
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
jaks
Ryan
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching