ACT Math Essential Formulas Summary | ACT 数学常用运算公式汇总

📚 ACT Math Essential Formulas Summary | ACT 数学常用运算公式汇总

The ACT Math section requires quick recall of a wide range of formulas across arithmetic, algebra, geometry, trigonometry, and statistics. This summary gathers the essential formulas you must know to solve problems efficiently and accurately. Memorize these, and you will save precious time on test day.

ACT 数学部分要求考生快速回忆涵盖算术、代数、几何、三角和统计的广泛公式。本汇总整理了必须掌握的核心公式，以便高效、准确地解题。熟记这些内容，你将在考试当日节省宝贵时间。

---

1. Arithmetic and Number Properties | 算术与数论性质

Percent Change: It measures the relative increase or decrease from an original value.

Percent Change = (New – Original) / |Original| x 100%

百分比变化：衡量相对于原值的增减幅度。

百分比变化 = (新值 – 原值) / |原值| x 100%

Arithmetic Mean (Average): The sum of all terms divided by the number of terms.

Mean = (Sum of terms) / (Number of terms)

算术平均数（均值）：所有项之和除以项数。

均值 = (各项之和) / (项数)

Distance-Rate-Time Relationship: Distance equals rate multiplied by time.

d = r x t

距离-速率-时间关系：距离等于速率乘以时间。

d = r x t

Greatest Common Factor (GCF) and Least Common Multiple (LCM): For two positive integers a and b, their product equals the product of their GCF and LCM.

a x b = GCF(a, b) x LCM(a, b)

最大公因数（GCF）与最小公倍数（LCM）：对于两个正整数 a 和 b，其乘积等于 GCF 与 LCM 的乘积。

a x b = GCF(a, b) x LCM(a, b)

Prime Factorization: Every integer greater than 1 can be expressed as a unique product of prime numbers. This is essential for simplifying fractions and finding LCM/GCF.

质因数分解：每个大于 1 的整数都可以唯一地表示为质数的乘积。这在化简分数和求 LCM/GCF 时至关重要。

---

2. Linear Equations and Inequalities | 线性方程与不等式

Slope Formula: The slope of a line passing through (x₁, y₁) and (x₂, y₂).

m = (y₂ – y₁) / (x₂ – x₁)

斜率公式：经过 (x₁, y₁) 和 (x₂, y₂) 的直线斜率。

m = (y₂ – y₁) / (x₂ – x₁)

Slope-Intercept Form: Equation of a line with slope m and y-intercept b.

y = mx + b

斜截式：斜率为 m、y 截距为 b 的直线方程。

y = mx + b

Point-Slope Form: Equation of a line given slope m and a point (x₁, y₁).

y – y₁ = m(x – x₁)

点斜式：给定斜率 m 和一点 (x₁, y₁) 的直线方程。

y – y₁ = m(x – x₁)

Midpoint Formula: The midpoint of the segment joining (x₁, y₁) and (x₂, y₂).

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)

中点公式：连接 (x₁, y₁) 和 (x₂, y₂) 的线段的中点。

中点 = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Inequality Rules: When multiplying or dividing both sides of an inequality by a negative number, reverse the inequality sign.

不等式规则：当不等式两边同时乘以或除以一个负数时，必须反转不等号方向。

---

3. Quadratic Equations and Functions | 二次方程与函数

Standard Form of a Quadratic: ax² + bx + c = 0, where a ≠ 0.

二次方程标准式：ax² + bx + c = 0，其中 a ≠ 0。

Quadratic Formula: Solutions to ax² + bx + c = 0.

x = (-b ± √(b² – 4ac)) / (2a)

二次求根公式：ax² + bx + c = 0 的解。

x = (-b ± √(b² – 4ac)) / (2a)

Discriminant: The expression under the square root, Δ = b² – 4ac, determines the nature of the roots (two real, one real, or two complex).

判别式：根号下的表达式 Δ = b² – 4ac 决定根的性质（两相异实根、一重根或两共轭复根）。

Vertex of a Parabola: For f(x) = ax² + bx + c, the x-coordinate of the vertex is h = -b/(2a), and the y-coordinate is k = f(h).

Vertex (h, k) = (-b/(2a), f(-b/(2a)))

抛物线顶点：对于 f(x) = ax² + bx + c，顶点 x 坐标为 h = -b/(2a)，y 坐标为 k = f(h)。

顶点 (h, k) = (-b/(2a), f(-b/(2a)))

Factoring Patterns: Recognizing difference of squares: a² – b² = (a + b)(a – b) and perfect square trinomials: a² ± 2ab + b² = (a ± b)².

因式分解模型：平方差公式 a² – b² = (a + b)(a – b)；完全平方三项式 a² ± 2ab + b² = (a ± b)²。

---

4. Exponents and Radicals | 指数与根式

Product of Powers: When multiplying like bases, add the exponents.

aᵐ x aⁿ = aᵐ⁺ⁿ

同底数幂的乘积：底数相同时，指数相加。

aᵐ x aⁿ = aᵐ⁺ⁿ

Power of a Power: Multiply the exponents.

(aᵐ)ⁿ = aᵐⁿ

幂的乘方：指数相乘。

(aᵐ)ⁿ = aᵐⁿ

Negative Exponent: A negative exponent indicates a reciprocal.

a⁻ⁿ = 1 / aⁿ

负指数：负指数表示倒数。

更多咨询请联系16621398022（同微信）