📚 Quadratic Functions: Graphs and Properties | 二次函数的图像与性质
A quadratic function is one of the most important topics in algebra. It is defined as a polynomial of degree two, and its graph is a smooth, symmetric curve called a parabola. Understanding its shape, vertex, axis of symmetry, and roots is essential for solving many real-world problems.
二次函数是代数中最重要的内容之一。它是一类最高次数为二的整式函数，其图像是一条平滑且对称的曲线，称为抛物线。理解它的形状、顶点、对称轴和根，对解决许多现实问题至关重要。
1. Standard Form and General Form | 标准形式与一般形式
The standard form of a quadratic function is written as:
二次函数的标准形式写作：
f(x) = ax² + bx + c, a ≠ 0
r. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).
其中 a、b、c 为实数。若 a = 0，函数就变成了一次函数。a 的正负决定抛物线开口方向：a > 0 时开口向上，a < 0 时开口向下。
A quadratic function can also be written in vertex form, which makes the vertex easy to read:
二次函数也可以写成顶点式，这样可以直观地看出顶点坐标：
f(x) = a(x − h)² + k, a ≠ 0
In vertex form, the vertex is exactly the point (h, k). The two forms are equivalent, and you can convert between them by completing the square.
在顶点式中，顶点就是点 (h, k)。这两种形式可以互相转换，常用的方法就是配方法。
2. Axis of Symmetry | 对称轴
Every parabola has a vertical line of symmetry that passes through its vertex. This line divides the curve into two mirror-image halves.
每条抛物线都有一条通过顶点的竖直对称轴，它把曲线分成左右完全对称的两部分。
For the standard form f(x) = ax² + bx + c, the axis of symmetry is given by:
对于标准形式 f(x) = ax² + bx + c，对称轴为：
x = −b / (2a)
For the vertex form f(x) = a(x − h)² + k, the axis is simply the line x = h.
对于顶点式 f(x) = a(x − h)² + k，对称轴就是直线 x = h。
Example: For f(x) = 2x² − 8x + 3, the axis of symmetry is x = −(−8) / (2 × 2) = 2.
例：对于 f(x) = 2x² − 8x + 3，对称轴为 x = −(−8) / (2 × 2) = 2。
3. Vertex and Extreme Values | 顶点与最值
The vertex is the highest or lowest point of the parabola. It represents the maximum or minimum value of the function.
顶点是抛物线的最高点或最低点，它对应函数的最大值或最小值。
- If a > 0, the parabola opens upward and the vertex is the minimum point.
- 如果 a > 0，抛物线开口向上，顶点是最低点。
- If a < 0, the parabola opens downward and the vertex is the maximum point.
- 如果 a < 0，抛物线开口向下，顶点是最高点。
vertex is x = −b / (2a). Substitute this value into the function to find the y-coordinate.
顶点的横坐标是 x = −b / (2a)，把它代入函数即可求出纵坐标。
Vertex = ( −b/(2a), f( −b/(2a) ) )
Example: = x² − 4x + 5, the vertex is at x = 2 and y = 1. Since a = 1 > 0, the minimum value is 1.
例：对于 f(x) = x² − 4x + 5，顶点为 x = 2，y = 1。因为 a = 1 > 0，所以最小值为 1。
4. Roots and the Discriminant | 根与判别式
alues where f(x) = 0. They correspond to the points where the parabola crosses the x-axis.
二次函数的根（也称为零点）是使 f(x) = 0 的 x 值，也就是抛物线与 x 轴交点的横坐标。
For ax² + bx + c = 0, the quadratic formula gives:
对于 ax² + bx + c = 0，求根公式给出：
x = ( −b ± √b² − 4ac ) / (2a)
The expression b² − 4ac is called the discriminant, usually denoted as Δ.
式子 b² − 4ac 称为判别式，通常用 Δ 表示。
| Δ = b² − 4ac | Number of Real Roots | 实数根个数 |
| Δ > 0 | Two distinct real roots | 两个不相等实数根 |
| Δ = 0 | One repeated real root | 两个相等实数根（一个重根） |
| Δ < 0 | No real roots | 无实数根 |
x-axis twice; if Δ = 0, it touches the x-axis once; if Δ < 0, it never touches the x-axis.
判别式还能告诉我们图像与 x 轴的相对位置。Δ > 0 时，抛物线与 x 轴有两个交点；Δ = 0 时，与 x 轴只有一个切点；Δ < 0 时，与 x 轴没有交点。
5. Transformations of the Parent Function | 二次函数的图像变换
pes of transformations are important: vertical shifts, horizontal shifts, and reflections.
所有二次函数都可以看作基本图像 y = x² 的变换结果。重要的变换有三种：上下平移、左右平移和反射。
- Vertical shift: f(x) = x² + k moves the graph up (k > 0) or down (k < 0).
- 上下平移：f(x) = x² + k 使图像向上（k > 0）或向下（k < 0）移动。
- Horizontal shift: f(x) = (x − h)² moves the graph right (h > 0) or left (h < 0).
- 左右平移：f(x) = (x − h)² 使图像向右（h > 0）或向左（h < 0）移动。
- Reflection and stretch: f(x) = ax² opens upward if a > 0, downward if a < 0. |a| > 1 makes the graph narrower; |a| < 1 makes it wider.
- 反射与伸缩：f(x) = ax² 中 a > 0 时开口向上，a < 0 时开口向下；|a| > 1 使图像更窄，|a| < 1 使图像更宽。
from y = x², horizontally shift by h, vertically stretch by a, then vertically shift by k.
在顶点式 f(x) = a(x − h)² + k 中，变换顺序为：从 y = x² 出发，先左右平移 h，再按 a 伸缩，最后上下平移 k。
6. Completing the Square | 配方法
Completing the square is a powerful technique used to convert standard form into vertex form.
配方法是一种将标准式转化为顶点式的常用技巧。
To convert f(x) = ax² + bx + c, follow these steps:
将 f(x) = ax² + bx + c 转化为顶点式，步骤如下：
- Factor out a from the first two terms: f(x) = a(x² + (b/a)x) + c
- 提取前两项的公因数 a：f(x) = a(x² + (b/a)x) + c
- Add and subtract (b/(2a))² inside the bracket.
- 在括号内加上并减去 (b/(2a))²。
- Factor the perfect square trinomial and simplify the constant term.
- 将完全平方三项式因式分解，并合并常数项。
Example: Convert f(x) = x² + 6x + 5.
例：将 f(x) = x² + 6x + 5 化为顶点式。
f(x) = (x² + 6x + 9) + 5 − 9 = (x + 3)² − 4
Thus the vertex is (−3, −4).
因此顶点为 (−3, −4)。
7. Graphing a Quadratic Function | 绘制二次函数图像
A quick and reliable way to graph a quadratic function involves five key points.
绘制二次函数图像时，一种快速且可靠的方法是先确定五个关键点。
| Step | Action | 操作 |
| 1 | Find the axis of symmetry x = −b/(2a) | 求对称轴 x = −b/(2a) |
| 2 | Find the vertex | 求顶点坐标 |
| 3 | Find the y-intercept: (0, c) | 求 y 轴截距：(0, c) |
| 4 | Find the x-intercepts by solving f(x) = 0 if possible | 解 f(x) = 0 求 x 轴截距（如果存在） |
| 5 | Use symmetry to plot the mirror point of the y-intercept | 利用对称性画出 y 轴截距的对称点 |
Once these points are plotted, draw a smooth curve through them. Always label the vertex and the axis of symmetry.
标出这些点之后，用平滑的曲线连接各点。记得标出顶点和对称轴。
8. Sign of Quadratic and Inequalities | 二次符号与不等式
The sign of a quadratic expression tells us when the parabola lies above or below the x-axis.
二次表达式的符号表示抛物线位于 x 轴上方还是下方。
If a > 0 and Δ > 0, the quadratic is positive outside the interval between the roots and negative between the roots:
如果 a > 0 且 Δ > 0，则二次函数在两根之外为正，在两根之间为负：
- x < r₁ or x > r₂: f(x) > 0
- x < r₁ 或 x > r₂ 时：f(x) > 0
- r₁ < x < r₂: f(x) < 0
- r₁ < x < r₂ 时：f(x) < 0
If a < 0, the inequalities are reversed. This concept is used to solve quadratic inequalities such as x² − 5x + 6 > 0.
如果 a < 0，不等号方向相反。这一概念常用于解二次不等式，如 x² − 5x + 6 > 0。
= 0 gives x = 2 and x = 3. Since a > 0, the solution to the inequality is x < 2 or x > 3.
解 x² − 5x + 6 = 0 得 x = 2 和 x = 3。因为 a > 0，所以不等式的解为 x < 2 或 x > 3。
9. Relationship Between Coefficients and Graph | 系数与图像的关系
Each coefficient in ax² + bx + c affects the graph in a specific way.
ax² + bx + c 中的每个系数都对图像有特定的影响。
- a determines the direction and width of the parabola.
- a 决定抛物线的开口方向和宽窄。
- b affects the position of the vertex along the x-axis, together with a.
- b 与 a 共同影响顶点在 x 方向上的位置。
- c is the y-intercept, where the graph crosses the y-axis.
- c 是 y 轴截距，即图像与 y 轴的交点。
Specifically, the vertex's x-coordinate is −b/(2a). If a and b have the same sign, the vertex lies to the left of the y-axis; if they have opposite signs, the vertex lies to the right.
具体来说，顶点的横坐标是 −b/(2a)。若 a 和 b 同号，顶点在 y 轴左侧；若异号，顶点在 y 轴右侧。
The sum and product of the roots appear in the coefficients:
根的和与积也体现在系数中：
r₁ + r₂ = −b/a, r₁ × r₂ = c/a
These relationships, known as Vieta's formulas, are frequently tested in exams.
这些关系称为韦达定理，是考试中的常考内容。
10. Applications and Problem Solving | 实际应用与解题
Quadratic functions appear in projectile motion, area optimization, economics, and physics. A common exam problem involves finding the maximum height of an object.
二次函数广泛出现在抛体运动、面积优化、经济学和物理学中。一个常见的考试题目是求物体上升的最大高度。
Suppose the height of a ball thrown upward is modeled by:
假设一个球被抛出后，其高度满足：
h(t) = −5t² + 20t + 1
where h is in meters and t is in seconds. The maximum height occurs at the vertex:
其中 h 的单位是米，t 的单位是秒。最大高度出现在顶点处：
t = −20 / (2 × (−5)) = 2 s, h(2) = −5(4) + 20(2) + 1 = 21 m
The ball reaches a maximum height of 21 meters after 2 seconds.
球在 2 秒时达到最大高度 21 米。
− x) = −x² + 20x. The maximum area occurs when x = 10 m, giving a square with area 100 m².
对于面积问题，若矩形周长为 40 m，则面积 A(x) = x(20 − x) = −x² + 20x。当 x = 10 m 时面积最大，此时为边长为 10 m 的正方形，面积为 100 m²。
11. Common Mistakes and Tips | 常见错误与备考提示
Students often make the following mistakes when working with quadratic functions.
学生在学习二次函数时，常犯以下错误。
- Forgetting that a cannot be zero.
- 忘记 a 不能为零。
- Using the wrong sign in the vertex formula: remember x = −b/(2a), not b/(2a).
- 顶点公式符号用错：注意是 x = −b/(2a)，不是 b/(2a)。
- the direction of horizontal shift in vertex form: f(x) = (x − h)² shifts right when h > 0.
- 在顶点式中弄错左右平移方向：f(x) = (x − h)² 中 h > 0 时右移。
- Forgetting to check whether a quadratic has real roots before trying to find x-intercepts.
- 求 x 轴截距前忘记先判断是否具有实数根。
- Mixing up the vertex coordinates when translating between forms.
- 在两种形式之间转换时，混淆顶点坐标。
To succeed, always write the function in a clear form, compute the discriminant first, and draw a rough sketch to verify your results.
要取得好成绩，应先把函数写成清晰的形式，先算判别式，再画简图验证结果。
12. Practice Checklist | 练习清单
Use this checklist to confirm you have mastered the key skills.
用以下清单检查你是否掌握了核心技能。
| Skill | 技能 | Mastered? | 掌握？ |
| Identify a, b, c from standard form | 从标准式中识别 a、b、c | ☐ | ☐ |
| Find the axis of symmetry and vertex | 求对称轴与顶点 | ☐ | ☐ |
| Sketch the parabola using 5 key points | 用五个关键点画抛物线 | ☐ | ☐ |
| Use the discriminant to describe roots | 用判别式判断根的情况 | ☐ | ☐ |
| Apply Vieta's formulas | 应用韦达定理 | ☐ | ☐ |
| Solve quadratic inequalities | 解二次不等式 | ☐ | ☐ |
Keep practicing graphing by hand, because exam questions often require you to interpret the graph rather than only compute formulas.
坚持手绘图像练习，因为考试题往往要求你解读图像，而不仅仅是套用公式。
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