Objectives (5 - 7 minutes)

1. Understand the concept of exponential function and its graphical representation.
2. Analyze and interpret the behavior of the graph of an exponential function.
3. Apply the acquired knowledge to solve practical problems involving exponential functions.

Secondary Objectives:

• Develop the ability to plot graphs of exponential functions on graph paper.
• Reinforce the understanding of previous mathematical concepts, such as exponentiation and numerical base.
• Stimulate critical thinking and problem-solving in a logical and analytical way.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher should start the lesson by briefly reviewing the concepts of exponentiation and exponential functions. It is important for students to understand these concepts before moving on to the study of exponential functions. The teacher can do this through simple examples and active student participation. (2 - 3 minutes)

2. Problem situations:

   • The teacher can present the following problem: 'Imagine we have a population of bacteria that doubles every hour. How can we mathematically represent this situation? What would be the graph of this function over time?' This problem serves to contextualize the importance of exponential functions and their graph. (3 - 4 minutes)
   • Another problem situation that the teacher can present is: 'If we have a sum of money invested in a bank that yields compound interest, how can we mathematically represent this situation? What would be the graph of this function over time?' This problem serves to show that exponential functions can be applied in different contexts, not just in biology. (3 - 4 minutes)

3. Contextualization: The teacher should explain that exponential functions are widely used in science, economics, engineering, and many other fields of everyday life. For example, in physics, exponential functions are often used to describe radioactive decay and population growth. In economics, exponential functions are used to model investment growth. (2 - 3 minutes)

4. Topic presentation: The teacher should introduce the topic of the lesson - Graphs of Exponential Functions. The teacher can say: 'Today, we will learn how to graphically represent the exponential functions we have studied. We will see how exponential function graphs look and what they tell us about the function's behavior.' (1 - 2 minutes)

Development (20 - 25 minutes)

1. Theory of Exponential Functions Graph (7 - 9 minutes)

   • The teacher should start by explaining that the graph of an exponential function is always a curve, never a straight line, because the function's growth rate is increasing.
   • Next, the teacher should emphasize that the graph of an increasing exponential function (when the base is greater than 1) approaches the x-axis but never crosses it. On the other hand, the graph of a decreasing exponential function (when the base is between 0 and 1) approaches the x-axis but never crosses it.
   • The teacher should emphasize that the slope of the graph of an exponential function increases as x increases, which means that the function's growth rate is increasing.
   • The teacher should explain that the base of the function determines the 'steepness' of the curve. For example, an exponential function with a base of 2 increases more rapidly than an exponential function with a base of 1.5.

2. Practical Examples (8 - 10 minutes)

   • The teacher should show several examples of exponential functions and their graphs, explaining step by step how to plot the graph. The teacher should use different bases (for example, 2, 10, 0.5) to illustrate how the base affects the graph.
   • The teacher should explain how to find the domain and range of the function from the graph. The teacher should emphasize that the domain of an exponential function is always the set of real numbers, and the range is the set of positive numbers.
   • The teacher should also explain how to find the image of the function from the graph. The image of an exponential function is the set of all values the function can take. The teacher should emphasize that the image of an increasing exponential function is the set of positive numbers, and the image of a decreasing exponential function is the set of positive numbers.

3. Problem Solving (5 - 6 minutes)

   • The teacher should present some problems involving exponential functions and ask students to solve them. The problems should involve interpreting graphs of exponential functions and applying the acquired knowledge to solve practical problems.
   • The teacher should circulate around the classroom, observing the students' work and offering assistance as needed.

By the end of this stage, students should be able to plot the graph of an exponential function, interpret the graph's behavior, and solve practical problems involving exponential functions.

Return (10 - 12 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should propose that students discuss in small groups about the solutions found for the proposed problems. Each group should present their conclusions to the class, explaining the reasoning used to reach the result.
   • During the discussion, the teacher should encourage the participation of all students, asking questions to verify understanding and clarify possible doubts. The teacher should ensure that all key concepts have been understood and that students feel comfortable expressing their ideas.

2. Connection with Theory (3 - 4 minutes)

   • After the group discussion, the teacher should make the connection between the solutions presented by the students and the theory studied. The teacher can ask: 'How did what we learned about the graph of exponential functions help us solve these problems?'.
   • The teacher should highlight the importance of understanding the behavior of the graph of an exponential function for solving practical problems. The teacher can use examples from the discussed problems to illustrate this point.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should propose that students reflect individually on what they learned in the lesson. The teacher can ask questions like: 'What was the most important concept you learned today?' and 'What questions have not been answered yet?'.
   • The teacher should give a minute for students to think about these questions. Then, the teacher can ask some students to share their answers with the class. The teacher should encourage students to be honest in their answers and not be afraid to express their doubts.

4. Feedback (2 - 3 minutes)

   • Finally, the teacher should ask for feedback from students about the lesson. The teacher can ask questions like: 'What did you find most interesting in today's lesson?' and 'What could be improved in the next lesson?'.
   • The teacher should take note of the students' answers and use them to plan future lessons. Student feedback is a valuable tool for improving teaching and learning.

By the end of this stage, students should have a clear understanding of what they learned in the lesson, be able to make connections between theory and practice, and feel comfortable expressing their doubts and suggestions.

Conclusion (3 - 5 minutes)

1. Summary of Contents (1 - 2 minutes)

   • The teacher should recap the main points covered during the lesson. This includes the concept of exponential function, the interpretation of its graph, and how to plot the graph of an exponential function.
   • The teacher should reinforce that the exponential function is a function that grows or decreases rapidly, depending on the base, and that its graph never touches the x-axis.
   • The teacher should remind students that the base of the exponential function determines the 'steepness' of the curve. A base greater than 1 results in an increasing exponential function, while a base between 0 and 1 results in a decreasing exponential function.

2. Connection between Theory, Practice, and Applications (1 - 2 minutes)

   • The teacher should highlight how the lesson connected theory, practice, and applications.
   • The teacher should emphasize that plotting the graph of an exponential function is an essential practical skill for interpreting and solving problems involving exponential functions.
   • The teacher should recall the practical applications of exponential functions, such as modeling population growth and compound interest.

3. Additional Materials (1 minute)

   • The teacher can suggest additional study materials for students who wish to deepen their knowledge of exponential functions and their graphs.
   • These materials may include explanatory videos, interactive math websites, textbooks, and online exercises.

4. Importance of the Subject (1 minute)

   • Finally, the teacher should emphasize the importance of exponential functions and their graphs in everyday life.
   • The teacher can mention again some of the practical applications, reinforcing that understanding these concepts is fundamental in various areas, such as science, economics, and engineering.
   • The teacher should encourage students to continue exploring and applying these concepts in their daily lives, and to see mathematics as a powerful and relevant tool.