FUNÇÃO AFIM | FUNÇÃO DO 1º GRAU | LEI DE FORMAÇÃO | AULA 1 - Professora Angela Matemática

Summary

In this lesson, the instructor introduces the concept of affine functions (also known as first-degree polynomial functions), explaining their key components: the coefficient of x (a) and the constant term (b). The video provides examples and real-life applications, such as how the cost of purchasing multiple items can be modeled with an affine function. The instructor also discusses the importance of the graphical representation of affine functions and their relevance in everyday scenarios. The lesson is geared toward helping students understand the practical uses of affine functions and how to express them mathematically.

Takeaways

• 😀 The lesson introduces the concept of linear functions (also known as first-degree polynomial functions or affine functions) and explains their mathematical structure.
• 😀 An affine function can be expressed as f(x) = ax + b, where 'a' and 'b' are real numbers, and 'a' cannot be zero.
• 😀 The number 'a' is called the angular coefficient, while 'b' is referred to as the linear coefficient or independent term.
• 😀 The script provides various examples of affine functions, including f(x) = 3x + 2, y = -5x + 1, and y = (1/3)x - 2, demonstrating how to identify 'a' and 'b'.
• 😀 The function y = 2x is given as a real-world example, showing that the cost of candies is directly proportional to the number of candies bought, with each candy costing 2 reais.
• 😀 The relationship between the number of candies and the cost is represented as an affine function, where 'f(x) = 2x' and the coefficient b is zero.
• 😀 The function can be graphically represented on a Cartesian plane, with points such as (1, 2), (2, 4), (3, 6), and (4, 8) corresponding to the number of candies and the total cost.
• 😀 The lesson emphasizes the significance of the angular coefficient ('a'), which determines the slope of the graph, and the linear coefficient ('b'), which determines the y-intercept.
• 😀 A second example involves a service provider charging for both a visit and the time spent at a client's house, where the total cost is represented by P = 40 + 20H (40 reais for the visit and 20 reais per hour).
• 😀 The instructor encourages viewers to take notes and stay engaged for further lessons, and mentions that more detailed lessons on graphing affine functions will be covered in future videos.

Q & A

What is a 'função afim' in mathematics?

A 'função afim' is a type of linear function, also known as a polynomial of the first degree or a first-degree function. It can be expressed in the form f(x) = ax + b, where 'a' is the coefficient of x (angular coefficient) and 'b' is the constant term (linear coefficient).

Why can't the coefficient 'a' be zero in a função afim?

If 'a' were zero, the equation would no longer be a linear function. Instead, it would become a constant function (f(x) = b), meaning the function would not depend on x, which violates the principle of a first-degree function.

What does the coefficient 'a' represent in the function f(x) = ax + b?

The coefficient 'a' represents the angular coefficient, which determines the slope or the rate of change of the function. It shows how much the function's output changes for a unit change in the input 'x'.

What does the coefficient 'b' represent in a função afim?

The coefficient 'b' is the linear coefficient, or the constant term. It represents the point where the function intersects the y-axis (the y-intercept) when x equals zero.

Can the coefficient 'b' be zero in a função afim?

Yes, 'b' can be zero. If 'b' equals zero, the function would still be a valid linear function, but it would pass through the origin (0,0) on the graph.

How can we interpret the function f(x) = 2x graphically?

The graph of f(x) = 2x is a straight line that passes through the origin (0,0) and has a slope of 2. This means for every unit increase in 'x', the output 'f(x)' increases by 2 units.

What does it mean when a função afim is said to be 'increasing'?

A função afim is increasing if the coefficient 'a' is positive, meaning as the input 'x' increases, the output 'f(x)' also increases. This is represented by an upward-sloping line on a graph.

How is a real-life example used to explain a função afim in the lesson?

In the lesson, a real-life example is used where the cost of buying sweets is modeled as a função afim. The cost is a function of the number of sweets purchased, with the cost per sweet being constant (2 reais per sweet), leading to the equation f(x) = 2x.

What is the key takeaway from the example of the service provider charging for visits and hours worked?

The example illustrates how a função afim can model the cost of services. The base charge is 40 reais (the constant term 'b'), and the variable charge is 20 reais per hour worked (the coefficient 'a'). The equation representing this situation is P = 40 + 20H.

What should be noted when interpreting the coefficients in the expression P = 40 + 20H?

In the expression P = 40 + 20H, '40' is the fixed fee (the coefficient 'b'), and '20' is the charge per hour worked (the coefficient 'a'). The value of 'H' represents the number of hours worked, and 'P' is the total amount paid.