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Questions and Answers

線性方程的標準形式是什么？

  • Ax + By = C (correct)
  • y - y1 = m(x - x1)
  • y = mx + b
  • A = P(1 + r/n)^(nt)

    • 圖形函數的用途是什么？

  • 識別函數的定義域和值域
  • 分析函數的行為
  • 所有上述 (correct)
  • 計算函數的根

    • 什麼是指數增長？

  • 增長速率的改變
  • 每年固定增長的百分比
  • 隨機增長的百分比
  • 在固定的時間間隔內增加的固定百分比 (correct)

    • 指數增長公式中的P是什么？

    <p>本金</p> Signup and view all the answers

    線性方程的圖形是什么？

    <p>直線</p> Signup and view all the answers

    什么是线性方程的斜率？

    <p>線的陡峭度</p> Signup and view all the answers

    Qué es una característica de crecimiento exponencial?

    <p>Una aceleración rápida en la tasa de crecimiento.</p> Signup and view all the answers

    ¿Cuál es la forma estándar de una función cuadrática?

    <p>f(x) = ax^2 + bx + c</p> Signup and view all the answers

    ¿Cuál es la fórmula para resolver ecuaciones cuadráticas?

    <p>x = (-b ± √(b^2 - 4ac)) / 2a</p> Signup and view all the answers

    ¿Qué es la operación de composición de funciones?

    <p>(f ∘ g)(x) = f(g(x))</p> Signup and view all the answers

    ¿Qué es el vértice de una parábola?

    <p>El punto más alto o más bajo de la parábola</p> Signup and view all the answers

    ¿Qué es la función de adición de funciones?

    <p>(f + g)(x) = f(x) + g(x)</p> Signup and view all the answers

    Study Notes

    Linear Equations

    • A linear equation is an equation in which the highest power of the variable(s) is 1.
    • Standard form: Ax + By = C, where A, B, and C are integers, and A and B are not both zero.
    • Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
    • Point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
    • Graphing linear equations:
      • The graph of a linear equation is a straight line.
      • The slope (m) determines the steepness and direction of the line.
      • The y-intercept (b) determines the point at which the line crosses the y-axis.

    Graphing Functions

    • The graph of a function is a visual representation of the relationship between the input (x) and output (y) values.
    • The graph can be used to:
      • Identify the domain and range of the function.
      • Determine the x-intercepts (roots) of the function.
      • Identify the y-intercept of the function.
      • Analyze the behavior of the function (increasing, decreasing, maxima, minima).

    Exponential Growth

    • Exponential growth occurs when a quantity increases by a fixed percentage at regular time intervals.
    • Exponential growth formula: A = P(1 + r/n)^(nt), where:
      • A is the amount of money accumulated after n years, including interest.
      • P is the principal amount (initial investment).
      • r is the annual interest rate (in decimal form).
      • n is the number of times that interest is compounded per year.
      • t is the time the money is invested or borrowed for, in years.
    • Characteristics of exponential growth:
      • Rapid acceleration in growth rate.
      • Eventual surpassing of linear growth.

    Quadratic Functions

    • A quadratic function is a polynomial function of degree two, meaning the highest power of the variable(s) is two.
    • Standard form: f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0.
    • Graphing quadratic functions:
      • The graph of a quadratic function is a parabola that opens upward or downward.
      • The vertex of the parabola is the minimum or maximum point of the function.
      • The axis of symmetry is the vertical line that passes through the vertex.
    • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, which solves quadratic equations of the form ax^2 + bx + c = 0.

    Function Operations

    • Function operations allow us to combine functions in different ways to create new functions.
    • Types of function operations:
      • Function addition: (f + g)(x) = f(x) + g(x)
      • Function subtraction: (f - g)(x) = f(x) - g(x)
      • Function multiplication: (f × g)(x) = f(x) × g(x)
      • Function division: (f ÷ g)(x) = f(x) ÷ g(x), as long as g(x) ≠ 0
    • Composition of functions: (f ∘ g)(x) = f(g(x)), where the output of g becomes the input for f.

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