Introduction - Exponential Function: Graph

Relevance of the Topic

Exponential functions are fundamental pillars in the study of mathematics, present in a wide variety of natural, physical, and economic phenomena. Understanding their graphs is essential to decipher growth and decay patterns, as well as to allow direct visualization of the rate of change of these processes, known as the rate of variation.

The graph of an exponential function, in turn, has characteristics that distinguish it from other types of functions, becoming a unique tool for understanding these processes. You know that image of a curve 'growing without limits' or 'decaying rapidly'? That's the graph of an exponential function! Understanding it will allow you to unravel various natural and artificial phenomena.

Contextualization

In the vast panorama of mathematics, the exponential function is found as a widely studied topic during high school. The study of the graphs of these functions fits into the curricular component of Functions and their Properties, being a natural extension of the study of the exponential function itself.

At this point, you should already be familiar with the basic concepts of a function and what an exponential function is. Now, let's delve into the visualization of these functions through their graphs. For this, we will use the knowledge base you already have about functions, Cartesian graphs, and basic properties of exponentiation.

Theoretical Development - Exponential Function: Graph

Components

• Coordinate axes: The graph of an exponential function is plotted on a mathematical structure known as the Cartesian plane or coordinate axes. This arrangement allows the simultaneous representation of domains and codomains, creating a visual image of the function.

• Formation law: The formation law of an exponential function is what actually determines the shape of its graph. For the exponential function in the generic form y = a^x, a is the base and x is the exponent. The value of a and the direction of the graph (growth or decay) are crucial elements for visual understanding.

• Domain and Codomain: In the exponential function, the domain (set of possible values for x) and the codomain (corresponding values for y) must be considered when plotting the graph. For example, if the base a is positive, the domain will encompass all real numbers, and if a is negative, the domain will be only for integer exponents.

• Notable points: In the graph of an exponential function, some points stand out, such as the origin (0,1) which is always present. Furthermore, the graph never touches the y-axis, but can approach it infinitely, depending on the value of a.

Key Terms

• Graph of an exponential function: It is the visual representation of the exponential function, showing the relationships between the values of x and y. It has a continuous curve shape that can be growing or decaying, depending on the base a.

• Asymptote: It is the straight line, in the graph of an exponential function, that the curve approaches as x moves away to positive or negative infinity on the horizontal axis. In the case of the exponential function, the asymptote is always the x-axis.

• Exponential growth and exponential decay: One of the most important concepts when it comes to exponential function graphs. If the base a is greater than 1, the function will exhibit exponential growth. If the base is a number between 0 and 1, we will have exponential decay.

Examples and Cases

• Exponential function with base 2: The function y = 2^x has a graph of exponential growth. The curve starts smoothly, but quickly becomes steep as x advances. It never touches the y-axis and approaches it infinitely.

• Exponential function with base 1/2: The function y = (1/2)^x has a graph of exponential decay. The curve starts with a sharp growth and approaches the x-axis without touching it as x increases.

• Exponential function with base -2: The function y = (-2)^x has a peculiar graph with sign alternation. At each step on the x-axis, the function alternates between a positive and a negative value, creating small oscillations in the graph.

Detailed Summary - Exponential Function: Graph

Key Points

• Graph components: The graph of an exponential function is formed by the interaction of the coordinate axes (Cartesian plane) and the function formation law, represented by y = a^x.

• Formation law: The formation law expresses the relationship between x and y in an exponential function, where a is the base and x is the exponent. The value of a is crucial to determine the graph's behavior.

• Domain and codomain: The domain and codomain of the exponential function (x and y respectively) are fundamental to understand the proportions and limits of growth or decrease in the graph. Depending on the value of a, the domain can be all real numbers (for positive a) or only integer exponents (for negative a).

• Notable points: In the graph of an exponential function, the origin (0,1) is always a notable point, as it represents the beginning of the relationship between x and y. Furthermore, the exponential function will never touch the y-axis, but can approach it infinitely.

• Asymptote: In the graph of an exponential function, the asymptote is the straight line to which the curve approaches but never touches. In the case of the exponential function, the asymptote is always the x-axis.

• Exponential growth and decay: The graph's behavior depends on the base a of the exponential function. If a is greater than 1, the graph will show exponential growth. If a is a number between 0 and 1, we will have exponential decay. For negative a, we will have sign oscillation.

Conclusions

• Visual interpretation: The ability to visually interpret the graph of an exponential function is an essential skill for understanding and analyzing various natural, physical, and economic phenomena.

• Relationship between base and graph: The value of the base a in an exponential function plays a significant role in the graph's formation, determining whether it will show exponential growth, exponential decay, or sign oscillation.

• Practical utility: Understanding the graph of an exponential function has several practical applications, such as in modeling everyday phenomena and predicting the growth or decrease of a quantity over time.

Exercises

1. Plot the graph of the exponential function y = 3^x

2. Identify the base and characteristics of the graph of the exponential function y = (1/4)^x

3. Given the graph of an exponential function, write the formation law of the function.