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Questions and Answers
線性方程的標準形式是什么？
圖形函數的用途是什么？
什麼是指數增長？
指數增長公式中的P是什么？
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線性方程的圖形是什么？
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什么是线性方程的斜率？
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Qué es una característica de crecimiento exponencial?
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¿Cuál es la forma estándar de una función cuadrática?
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¿Cuál es la fórmula para resolver ecuaciones cuadráticas?
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¿Qué es la operación de composición de funciones?
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¿Qué es el vértice de una parábola?
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¿Qué es la función de adición de funciones?
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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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