## Determine if a table represents a linear or exponential

36 = y As can be shown, the function f (x) = 4 (3) x satisfies the conditions of being exponential and containing the point (2, 36), making it the correct choice. Spaniard: f(x) = 4(3)x = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) = f(x) Step-by-step explanation: To see if the point (2, 36) belongs to any of the described essential functions, we evaluated them: f(x) = 4(3)x f(x) = y = 36 y x = 2 f(x) = y = 36 y 4*32y = 4*9y = 36 Evaluamosy As can be shown, the function f(x) = 4(3)x meets the conditions of being exponencial and containing the point (2, 36), making it the correct alternative.

## Writing an exponential function given 2 points.

We were given an exponential function in the previous cases, which we then evaluated for a given input. We are often given knowledge about an exponential function without actually understanding what the function is. We must use the data to write the function’s type, then specify the constants a and b, and finally evaluate the function.
80 deer were released into a wildlife refuge in 2006. By 2012, the herd had swelled to 180 deer. The population was rapidly increasing. Create an algebraic function N(t) to represent the N deer population over time t.
The number of years after 2006 will be our independent variable t. As a result, the problem’s data can be written as input-output pairs: (0, 80) and (6, 180). We have given ourselves the initial value for the equation, a = 80, by choosing our input variable to be calculated as years after 2006. We can now find b by plugging the second point into the equation [latex]Nleft(tright)=80bt[/latex]:

### Ex 1: determine if a table represents a linear or exponential

We were either given a function to graph or test directly in previous parts of this chapter, or we were given a set of points that were guaranteed to lie on the curve. After that, we used algebra to find the equation that perfectly matched the points. In this section, we use regression analysis, a modeling technique, to find a curve that models data from real-world observations. We don’t consider any of the points on the curve to be perfectly aligned when performing regression analysis. The goal is to find the best model for the data. The model is then used to make predictions about what will happen in the future.
The term “model” should not be misunderstood. Even though each has its own formal description, we sometimes use the terms function, equation, and model interchangeably in mathematics. The term “model” usually refers to an equation or function that approximates a real-world situation.
In this section, we’ll look at three different types of regression models: exponential, logarithmic, and logistic. We have a benefit because we have experience with each of these roles. Knowing their formal concepts, graph actions, and some of their real-world applications allows one to gain a better understanding of them. Key features and descriptions of the related function are provided with each regression model for analysis. Consider rethinking each of these features, reflecting on our previous work, and then investigating how regression is used to model real-world phenomena.

### Ex: find an exponential function given two points – initial

With a population of (1.353) billion people in 2018, India is the world’s second most populous country. The population is increasing at a rate of around (1% ) each year. If current trends continue, India’s population will surpass that of China by 2024. When populations grow exponentially, we sometimes refer to it as “exponential growth,” which means that everything is expanding at a breakneck pace. The word exponential growth, on the other hand, has a very particular definition for a mathematician. We’ll look at exponential functions in this section to see how they model rapid growth. In the Chapter 5.3 assignment in WeBWorK, you must solve problems involving exponential functions.
When we looked at linear growth, we noticed a constant rate of change, or how much the output increased with each unit increase in input. In the equation (f(x)=3x+4), for example, the slope indicates that the output increases by (3) each time the input increases by (x) (1\). Since we have a percent change per unit time (rather than a constant change) in the number of people in the India population example, the scenario is different.