Objectives (5 minutes)

1. Introduce the Concept of Quadratic Functions: The teacher will introduce the concept of quadratic functions, explaining that these are functions represented by a parabola and have a squared term as the highest power of the variable. The students are expected to understand what a quadratic function is and its representation as a parabola.

2. Understand the Components of Quadratic Functions: The teacher will discuss the standard form of quadratic functions (ax²+bx+c=0) and the components of quadratic functions, including the coefficient of the square term (a), the coefficient of the linear term (b), and the constant term (c). The students are expected to identify the components of a quadratic function and understand how these components affect the shape and position of the parabola.

3. Learn to Plot Quadratic Functions: The teacher will demonstrate how to plot a quadratic function on a graph. The students are expected to learn how to plot a quadratic function by finding the vertex, axis of symmetry, and intercepts, and sketching the parabola.

Secondary Objective: 4. Solve Quadratic Functions: If time permits, the teacher will introduce the concept of solving quadratic functions by factoring, completing the square, and using the quadratic formula. The students are expected to learn how to use different methods to solve quadratic functions and understand when to use each method.

Introduction (10 - 15 minutes)

1. Review of Previous Concepts: The teacher will begin the class by reminding the students of the basic concepts of functions and graphs, which they have previously studied. These concepts include the definition of a function, the x and y-axes, plotting points on a graph, and the concept of variables and coefficients.

2. Problem Situation 1: To introduce quadratic functions, the teacher presents a problem situation: "Suppose you throw a ball straight up in the air with a certain force. Can you predict the path of the ball?" This question is designed to make students think about how the ball's path would look like a curve, leading to the concept of a parabola.

3. Problem Situation 2: Another problem situation could be: "Consider a business that produces and sells a product. The profit made by the business depends on the number of products sold. The more products sold, the higher the profit, up to a certain point. After that point, producing and selling more products actually decreases the profit. Can you envision how the graph of profit versus the number of products sold would look?" This problem situation introduces the concept of a maximum point in a parabola.

4. Real-World Applications: The teacher will explain that quadratic functions are not just abstract mathematical concepts, but have practical applications in various fields like physics, economics, and engineering. For instance, in physics, the motion of objects thrown in the air can be modeled using quadratic functions. In economics, quadratic functions can be used to model profit maximization.

5. Introduction of Quadratic Functions: The teacher will then formally introduce the topic. "Today, we are going to study a special type of function called a quadratic function. A quadratic function can be represented graphically as a curve called a parabola. It is characterized by the highest power of its variable being 2."

6. Curiosity 1: The teacher could share that the concept of quadratic functions dates back to ancient times, with the Babylonians solving quadratic equations over 4000 years ago!

7. Curiosity 2: Another interesting fact the teacher could share is that when you graph a quadratic function, the curve you get, the parabola, has a unique property: any point on the parabola is equidistant from a fixed point (the focus) and a fixed line (the directrix).

By the end of the introduction, the students should be curious and motivated to learn more about quadratic functions. They should also understand the relevance of quadratic functions in real-world situations.

Development (20 - 25 minutes)

1. Introduction to Quadratic Functions (5 minutes)

   • The teacher initiates the lesson with the general form of quadratic functions, y = ax² + bx + c, where a, b, and c are constants and a is not equal to zero.
   • The teacher explains that the graph of a quadratic function is a parabola. If the coefficient 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downward.
   • Particular attention is given to the point that the highest degree of the function is 2, which defines it as a quadratic function.

2. Components of Quadratic Functions (5 minutes)

   • The teacher introduces the concept of the leading coefficient ('a') of the quadratic function, elucidating how this affects the direction of the parabola's opening.
   • The teacher continues by noting that the term with the 'b' coefficient is the linear term, and 'c' is a constant term that affects the position of the parabola.
   • The teacher further explains that the vertex of the parabola can be found using the formula (-b/2a, f(-b/2a)), and the axis of symmetry is the vertical line x = -b/2a.

3. Graphing Quadratic Functions (10 minutes)

   • Step 1: First, the teacher demonstrates on the board the process of plotting a quadratic function by starting with an example of a quadratic function.
   • Step 2: The teacher elucidates the steps required to find the vertex of the function using the formula above.
   • Step 3: Next, the teacher illustrates how to find the axis of symmetry based upon the x-coordinate of the vertex.
   • Step 4: Afterward, the teacher shows how to find the y-intercept of the parabola (which is simply the value of 'c' in the quadratic formula).
   • Step 5: Following this, the teacher explains how to find the x-intercepts (the roots, if they exist) by setting y = 0 and solving the resulting quadratic equation for x.
   • Step 6: Using all the calculated points (vertex, y-intercept, and x-intercepts if exist), the teacher plots the quadratic functions on the board and helps students understand how they form the shape of a parabola.

4. Practicing Quadratic Functions (5 minutes)

   • Step 1: The teacher hands out practice problems for the students to attempt in class.
   • Step 2: Students work on plotting functions on a graph on their own, and the teacher walks around the class checking and offering guidance where necessary.
   • Step 3: After the students have tried the problems, the teacher chooses some functions and asks students to explain their logic as they plot the function.

5. Solving Quadratic Functions (if time permits)

   • The teacher introduces the zero-factor property, demonstrating how to solve quadratic equations by factoring when it is possible.
   • The teacher also introduces the quadratic formula, explaining how it can be derived from the general quadratic equation by completing the square, and show how it can be used to find the roots of any quadratic equation.

Successively, the students should understand the concept of a quadratic function, prompt to identify and compute components of a quadratic function, and be familiar with plotting quadratic functions on a graph. These processes release students to understand and work with quadratic functions independently.

Feedback (10 minutes)

1. Connection with Real-Life Applications (3 minutes)

   • The teacher initiates the feedback phase by drawing connections between the concepts learned and their real-world applications. Examples include:
     • Physics: The teacher reinforces the idea that the path of a projectile follows a parabolic path, which can be modeled using a quadratic function.
     • Economics: The teacher reiterates the concept of profit maximization in businesses, which can be modeled using a quadratic function to find the optimal number of units to sell for maximum profit.
   • The teacher emphasizes that quadratic functions are not just mathematical concepts but tools that can be used to solve problems in various fields.

2. Reflection and Discussion (3 minutes)

   • The teacher encourages students to reflect on what they have learned in the lesson. The students are asked to consider the following questions:
     1. What was the most important concept learned today?
     2. What questions remain unanswered?
   • The students are given a couple of minutes to reflect on these questions before sharing their thoughts. The teacher can choose a few students to share their reflections with the class.

3. Assessment of Understanding (4 minutes)

   • The teacher can assess the students' understanding of the concepts covered in the lesson through a short quiz or exit ticket. This could include a few questions related to identifying and graphing quadratic functions and solving quadratic equations.
   • The teacher collects the exit tickets at the end of the class and reviews them to gauge the students' understanding of the topic. This will also help the teacher plan for any necessary review or reteaching in the next class.

By the end of the feedback session, the students should have a clear understanding of the practical applications of quadratic functions and be able to connect the theoretical concepts learned in class with real-world situations. They should also have an understanding of their own learning through reflection. The teacher, too, should have gained insights into the students' understanding of the topic, which can guide future instruction.

Conclusion (5 minutes)

1. Summarize and Recap: The teacher summarizes the main concepts learnt in the lesson.

   • The teacher reiterates the definition of a quadratic function, reminding students that these are functions that can be represented by a parabola, characterized by a squared term as the highest power of the variable.
   • The teacher reviews the components of a quadratic function (ax²+bx+c=0), including the coefficient of the square term (a), the coefficient of the linear term (b), and the constant term (c), and how these components affect the shape and position of the parabola.
   • The teacher re-emphasizes the process of plotting a quadratic function on a graph, including finding the vertex, axis of symmetry, and intercepts, and sketching the parabola.
   • If time allowed for the solving of quadratic functions, the teacher briefly revisits the methods used (factoring, completing the square, and the quadratic formula), and when to use each method.

2. Connecting Theory, Practice, and Applications: The teacher explains how the lesson connected theory to practice and real-world applications.

   • The teacher reminds the students that they applied the theoretical concept of quadratic functions in practice through graphing and solving quadratic functions.
   • The teacher highlights again that quadratic functions have practical applications in various fields like physics, economics, and engineering. For instance, the motion of objects thrown in the air can be modeled using quadratic functions, and profit maximization in businesses can be modeled using a quadratic function.

3. Additional Materials: The teacher provides suggestions for additional materials for study to complement the students' understanding of quadratic functions.

   • The teacher can recommend relevant chapters in the textbook or online resources such as Khan Academy, which provides video tutorials and practice exercises on quadratic functions.
   • The teacher can also suggest interactive online tools for graphing quadratic functions, such as Desmos, which can help students visualize and better understand the shape and characteristics of the parabola.

4. Importance of Quadratic Functions: Lastly, the teacher highlights the importance of understanding quadratic functions.

   • The teacher emphasizes that quadratic functions are not only fundamental in mathematics and essential for their further study in higher-level math courses, but also play a critical role in various real-world applications.
   • The teacher stresses that understanding quadratic functions can help students solve practical problems in their everyday life and future careers, especially if they choose a path in science, technology, engineering, or economics.

By the end of the conclusion, the students should have a clear understanding of what they learned during the lesson, how it connects to real-world applications, and what they can do to further study and practice the topic. They should also appreciate the importance and practicality of understanding quadratic functions.