## Questions and Answers

As equações lineares são expressões ______ que representam uma relação entre uma variável independente e uma variável dependente quando o gráfico da equação consiste de linhas retas.

algebraicas

Para resolver uma equação linear, você precisa isolar a variável de um ______ da equação.

lado

As equações lineares são usadas em várias aplicações em muitos campos, incluindo ______, física e engenharia.

economia

O objetivo ao resolver uma equação linear é terminar com uma equação que envolva apenas a sua variável e alguns ______.

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As funções quadráticas envolvem equações do segundo grau, representadas pela forma geral ______.

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Para calcular a mudança percentual, você divide a mudança no valor original pelo ______ e multiplica por 100.

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Enquanto as equações lineares são caracterizadas por linhas retas, as funções ______ têm curvas parabólicas.

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As funções quadráticas têm a forma geral de 'ax^2 + bx + c = 0', onde 'a', 'b' e 'c' são ______.

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As curvas parabólicas têm várias características-chave, como o vértice, que ocorre no ponto onde a função é ______ ou ______ dependendo se 'a' é positivo ou negativo, respectivamente.

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A modelagem com funções quadráticas é frequentemente usada para modelar fenômenos do mundo real, como receita ou a trajetória de ______.

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Para determinar a receita máxima (y) de um serviço de assinatura de jornal, você encontraria o vértice da equação quadrática, que representa o ______ ou ______.

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Entender os fundamentos das equações lineares e funções quadráticas é essencial para a modelagem matemática ______.

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## Study Notes

## Mathematical Modeling: Linear Equations and Quadratic Functions

Mathematical modeling involves using mathematical concepts and tools to understand and predict real-world phenomena. This can include everything from analyzing complex systems like the weather to determining optimal strategies for businesses and governments. Two types of functions commonly used in mathematical modeling are linear equations and quadratic functions. In this article, we will explore these two topics and their applications in problem solving.

## Linear Equations

Linear equations are algebraic expressions that represent a relationship between one independent variable and one dependent variable when the graph of the equation consists of straight lines. They have the form `ax + b = 0`

, where 'a' and 'b' are constants and 'x' represents the input.

### Solving Linear Equations

To solve a linear equation, you need to isolate the variable on one side of the equation. This can involve adding and subtracting the same value from both sides of the equation, multiplying or dividing both sides by a constant, or using algebraic properties to eliminate terms. The goal is to end up with an equation that only involves your variable and some constants.

### Applications of Linear Equations

Linear equations are used in various applications across many fields, including economics, physics, and engineering. For example, they can help determine the cost of goods based on quantity, predict the motion of objects under constant acceleration, and analyze supply and demand relationships in markets.

## Quadratic Functions

Quadratic functions are polynomial expressions of degree two, which means they consist of terms where the highest power of the independent variable is equal to two. They have the form `ax^2 + bx + c = 0`

, where 'a', 'b', and 'c' are constants. When graphed, quadratic functions result in curves known as parabolas.

### Characteristics of Parabolas

Parabolic curves have several key characteristics. The vertex, or highest point on the curve, occurs at the point where the function is minimized or maximized depending on whether 'a' is positive or negative respectively. Additionally, the axis of symmetry passes through this point and divides the curve into two identical halves.

### Modeling with Quadratic Functions

Quadratic functions are often used to model real-world phenomena such as revenue or the trajectory of projectiles. In these cases, you would write a quadratic equation for the situation, find the vertex of the equation (which represents the maximum or minimum), and then evaluate the equation at the x-coordinate of the vertex to get the corresponding y-value(s).

For instance, if you were modeling the relationship between the number of subscribers (x) and the price (y) of a newspaper subscription service, you could use a quadratic equation like `y = ax^2 + bx + c`

to represent the situation. By finding the vertex of this equation, you can determine the price (y) that maximizes revenue, assuming subscriptions are linearly related to the price.

## Linear vs Quadratic Functions

While linear equations are characterized by straight lines, quadratic functions have parabolic curves. This difference in shape allows quadratic models to capture more complex relationships than linear ones. However, both types of functions play crucial roles in mathematical modeling and problem solving across various disciplines.

In conclusion, understanding the basics of linear equations and quadratic functions is essential for effective mathematical modeling. These tools allow us to analyze patterns, predict outcomes, and make informed decisions in diverse fields ranging from finance to physics.

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