ESTUDO DAS FUNÇÕES: FUNÇÃO AFIM - PARTE I ( AULA 3 DE 9 ) | MATEMÁTICA INTEGRAL

Summary

In this video, Professor Alder Júnior explains the concept of affine functions (first-degree polynomials) in mathematics. He outlines the form of these functions, f(x) = ax + b, where 'a' is the coefficient of x and 'b' is a constant term. The video covers real-life examples, such as the salary of a real estate agent, to demonstrate how affine functions apply in practical scenarios. Additionally, it introduces linear functions (a special case where b = 0), the identity function, and how to graph affine functions. The content is designed to help students understand these fundamental mathematical concepts.

Takeaways

• 😀 Functions are essential mathematical concepts, and in this video, the focus is on affine functions, also called first-degree polynomial functions.
• 😀 An affine function has the form f(x) = ax + b, where 'a' is a real number and cannot be zero, while 'b' is a constant.
• 😀 The coefficient 'a' represents the slope of the line in an affine function, and 'b' is the y-intercept.
• 😀 A first-degree polynomial function is often referred to as an affine function, not a function of the first degree, as the term 'function' itself does not have a degree.
• 😀 A real-world example is given involving a real estate agent's salary, which consists of a fixed part (700 reais) and a variable part (2% of the sales made). This situation can be modeled with an affine function.
• 😀 The function representing the salary is f(x) = 700 + (2/100) * x, where 'x' is the value of sales for the month.
• 😀 If the total sales for the month are 300,000 reais, the agent's salary would be calculated by substituting x = 300,000 into the function, resulting in a total of 6,700 reais.
• 😀 A linear function is a special case of an affine function where b = 0. In this case, the function is represented as f(x) = ax.
• 😀 A function like f(x) = -4x is a linear function with a = -4 and b = 0.
• 😀 The identity function is a specific type of linear function where f(x) = x, meaning the output is always equal to the input. This function has a = 1 and b = 0.
• 😀 The graph of an affine function is a straight line, and if the function is constant (i.e., a = 0), the graph will be a horizontal line parallel to the x-axis, touching the y-axis at the point (0, b).

Q & A

What is the definition of an affine function?

An affine function, or first-degree polynomial function, is a function defined by the formula f(x) = ax + b, where 'a' and 'b' are real numbers, and 'a' cannot be zero.

Why must the coefficient 'a' in an affine function be different from zero?

The coefficient 'a' must be non-zero because if 'a' were zero, the function would no longer be a first-degree polynomial, and it would not have the characteristic of an affine function.

What does the term 'constant' refer to in an affine function?

The term 'constant' refers to the term 'b' in the affine function f(x) = ax + b. It is a value that does not change, irrespective of the value of 'x'.

What is the difference between an affine function and a linear function?

A linear function is a special case of an affine function where the constant term 'b' equals zero. In this case, the function takes the form f(x) = ax, where 'a' is the coefficient of 'x'.

What is a real-life example of an affine function mentioned in the video?

A real-life example of an affine function is the salary of a real estate agent, which consists of a fixed part (700 reais) and a variable part that depends on 2% of the value of the sales made that month.

How can the salary of a real estate agent be expressed as an affine function?

The salary of a real estate agent can be expressed as the function f(x) = 700 + (2/100) * x, where 'x' is the value of the sales made, and '700' is the fixed part of the salary.

If a real estate agent's sales total 300,000 reais in a month, how can we calculate their salary using the affine function?

To calculate the salary, substitute x = 300,000 into the function f(x) = 700 + (2/100) * x. This gives f(300,000) = 700 + (2/100) * 300,000, which results in 6,700 reais.

What is the graphical representation of an affine function?

The graph of an affine function is a straight line that is oblique to the x and y axes. The slope of the line is determined by the coefficient 'a', and the y-intercept is determined by the constant 'b'.

What is the function identity, and how is it related to affine functions?

The function identity is a special case of a linear function where f(x) = x. It is called the identity function because the output is always equal to the input, i.e., for any value of x, f(x) = x.

What happens when the coefficient 'a' in an affine function is zero?

When 'a' equals zero, the function becomes a constant function, where f(x) = b. In this case, the graph is a horizontal line that does not vary with 'x' and intersects the y-axis at the value of 'b'.