📚 Quadratic Functions: Graphs and Properties | 二次函数图像与性质
Quadratic functions are among the most fundamental topics in algebra. Their parabolic graphs appear throughout mathematics, physics, and engineering. Mastering the graph and properties of a quadratic function is essential for solving equations, inequalities, and optimization problems.
二次函数是代数中最基础的主题之一，其抛物线图像贯穿数学、物理和工程领域。掌握二次函数的图像与性质，对于求解方程、不等式和最优化问题至关重要。
1. Standard Form of a Quadratic Function | 二次函数的标准形式
A quadratic function is a polynomial of degree 2. Its general form is written as f(x) = ax² + bx + c, where a, b, and c are real numbers, and a ≠ 0.
二次函数是次数为2的多项式函数，其一般形式写作 f(x) = ax² + bx + c，其中 a、b、c 为实数，且 a ≠ 0。
The coefficient a determines the direction of the parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. The constants b and c affect the position of the graph but not its overall shape.
系数 a 决定抛物线的开口方向。当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。常数 b 和 c 影响图像的位置，但不改变其整体形状。
2. The Parabolic Shape | 抛物线的形状
The graph of any quadratic function is a smooth, symmetric curve called a parabola. Every parabola has exactly one turning point, known as the vertex, and is symmetric about a vertical line passing through this vertex.
任何二次函数的图像都是一条平滑且对称的曲线，称为抛物线。每条抛物线只有一个转折点，称为顶点，并且关于通过该顶点的一条竖直直线对称。
The axis of symmetry divides the parabola into two mirror-image halves. This symmetry allows us to plot only half of the points and then reflect them to complete the graph quickly.
对称轴将抛物线分为两个镜像对称的部分。这种对称性使我们只需描出部分点，再通过反射即可快速完成图像。
3. Vertex Form | 顶点式
The vertex form of a quadratic function is f(x) = a(x − h)² + k, where (h, k) is the vertex of the parabola. This form is especially useful because it directly reveals the coordinates of the turning point.
二次函数的顶点式为 f(x) = a(x − h)² + k，其中 (h, k) 是抛物线的顶点坐标。这种形式特别有用，因为它直接给出了转折点的坐标。
To convert from standard form to vertex form, we use the method of completing the square. For example, f(x) = x² − 4x + 5 can be rewritten as f(x) = (x − 2)² + 1, revealing a vertex at (2, 1).
要将标准形式转化为顶点式，我们使用配方法。例如，f(x) = x² − 4x + 5 可改写为 f(x) = (x − 2)² + 1，从而得到顶点坐标为 (2, 1)。
4. The Vertex and Axis of Symmetry | 顶点与对称轴
For a quadratic function in standard form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = −b / (2a). Substituting this value back into the function gives the y-coordinate, k.
对于标准形式的二次函数 f(x) = ax² + bx + c，其顶点的横坐标由 x = −b / (2a) 给出。将该值代回函数，即可得到纵坐标 k。
Vertex: ( −b/(2a), f(−b/(2a)) )
The axis of symmetry is the vertical line x = −b/(2a). This line passes through the vertex and is the mirror line for the parabola.
对称轴是竖直线 x = −b/(2a)。该直线经过顶点，是抛物线的镜像对称线。
5. Effect of Coefficient a | 系数 a 的影响
The coefficient a controls both the direction and the "width" of the parabola. When |a| is large, the parabola is narrow and steep; when |a| is small, it is wide and flat.
系数 a 同时控制抛物线的开口方向和“宽窄”。当 |a| 较大时，抛物线窄而陡；当 |a| 较小时，抛物线宽而平缓。
If a > 0: the parabola opens upward, and the vertex is a minimum point.
若 a > 0：抛物线开口向上，顶点为最小值点。
If a < 0: the parabola opens downward, and the vertex is a maximum point.
若 a < 0：抛物线开口向下，顶点为最大值点。
If a and b have the same sign, the vertex lies to the left of the y-axis; if opposite signs, to the right.
若 a 与 b 同号，顶点位于 y 轴左侧；若异号，则位于右侧。
6. Effect of Coefficient b | 系数 b 的影响
The coefficient b affects the horizontal position of the vertex. Changing b while keeping a and c constant shifts the parabola left or right while preserving its shape.
系数 b 影响顶点的水平位置。在保持 a 和 c 不变的情况下改变 b，会使抛物线左右平移，同时保持形状不变。
Algebraically, the axis of symmetry x = −b/(2a) depends directly on b. A larger positive b shifts the axis further to the left when a > 0, and further to the right when a < 0.
从代数角度看，对称轴 x = −b/(2a) 直接取决于 b。当 a > 0 时，b 的绝对值越大，对称轴越向左移；当 a < 0 时则越向右移。
7. Effect of Coefficient c | 系数 c 的影响
The constant term c represents the y-intercept of the parabola, i.e., the point where the graph crosses the y-axis at (0, c).
常数项 c 表示抛物线的 y 轴截距，即图像与 y 轴交于点 (0, c)。
Changing c shifts the entire graph vertically. Increasing c moves the parabola upward, while decreasing c moves it downward. This vertical shift does not affect the axis of symmetry.
改变 c 会使整个图像沿竖直方向平移。增大 c 使抛物线上移，减小 c 则使其下移。这种竖直平移不会改变对称轴的位置。
8. Discriminant and x-Intercepts | 判别式与 x 轴交点
The x-intercepts of a quadratic function are found by setting f(x) = 0 and solving the equation ax² + bx + c = 0. The number of real roots is determined by the discriminant Δ = b² − 4ac.
二次函数的 x 轴交点通过令 f(x) = 0 并求解方程 ax² + bx + c = 0 获得。实数根的个数由判别式 Δ = b² − 4ac 决定。
| Discriminant Δ | Number of x-intercepts | Graph Description |
| Δ > 0 | Two distinct roots | Parabola crosses the x-axis at two points |
| Δ = 0 | One repeated root | Parabola touches the x-axis at the vertex |
| Δ < 0 | No real roots | Parabola does not intersect the x-axis |
When Δ < 0, the graph lies entirely above the x-axis if a > 0, or entirely below if a < 0.
当 Δ < 0 时，若 a > 0，图像完全位于 x 轴上方；若 a < 0，则完全位于 x 轴下方。
9. Transformations of the Parabola | 抛物线的变换
Starting from the basic function f(x) = x², various transformations can be applied to obtain any parabola:
从基本函数 f(x) = x² 出发，可以通过各种变换得到任意抛物线：
Vertical shift: f(x) = x² + k moves the graph up (k > 0) or down (k < 0).
竖直平移：f(x) = x² + k 使图像上移（k > 0）或下移（k < 0）。
Horizontal shift: f(x) = (x − h)² moves the graph right (h > 0) or left (h < 0).
水平平移：f(x) = (x − h)² 使图像右移（h > 0）或左移（h < 0）。
Vertical stretch/compression: f(x) = ax² stretches if |a| > 1 and compresses if 0 < |a| < 1.
竖直伸缩：f(x) = ax² 在 |a| > 1 时拉伸，在 0 < |a| < 1 时压缩。
Reflection: f(x) = −x² reflects the graph across the x-axis.
翻折：f(x) = −x² 将图像关于 x 轴翻折。
10. Sketching the Graph: A Step-by-Step Guide | 绘制图像：分步指南
To sketch the graph of a quadratic function accurately, follow these steps:
要准确绘制二次函数的图像，请按以下步骤进行：
Identify the direction of opening from the sign of a.
根据 a 的符号确定开口方向。
Find the vertex using x = −b/(2a) and calculate the y-coordinate.
利用 x = −b/(2a) 求出顶点，并计算其纵坐标。
Determine the y-intercept at (0, c).
确定 y 轴截距 (0, c)。
Solve ax² + bx + c = 0 to find the x-intercepts, if they exist.
求解 ax² + bx + c = 0，若存在则求出 x 轴交点。
Plot the vertex, intercepts, and a few additional symmetric points, then draw a smooth curve.
描出顶点、交点及若干对称点，然后用平滑曲线连接。
11. Applications in Problem Solving | 在解题中的应用
Quadratic functions model many real-world scenarios, such as projectile motion, area optimization, and profit maximization. The vertex often represents a maximum or minimum value in these contexts.
二次函数可建模许多实际情境，如抛体运动、面积优化和利润最大化。在这些问题中，顶点通常代表最大值或最小值。
For example, when a ball is thrown upward, its height h(t) follows h(t) = −gt² + v₀t + h₀, where g is gravitational acceleration, v₀ is initial velocity, and h₀ is initial height. The maximum height occurs at t = −v₀/(2(−g)) = v₀/(2g).
例如，当球被向上抛出时，其高度 h(t) 满足 h(t) = −gt² + v₀t + h₀，其中 g 为重力加速度，v₀ 为初速度，h₀ 为初始高度。最大高度出现在 t = −v₀/(2(−g)) = v₀/(2g) 时刻。
Maximum height: hmax = v₀²/(2g) + h₀
Understanding the graph of a quadratic function allows us to interpret such problems geometrically and find solutions efficiently.
理解二次函数的图像，使我们能够从几何角度解读此类问题，并高效地找到解答。
12. Common Exam Pitfalls | 常见考试误区
Students often make avoidable mistakes when working with quadratic graphs. Being aware of these can help you earn full marks:
学生在处理二次函数图像时经常会犯一些本可避免的错误。注意以下几点，有助于你在考试中拿满分：
Forgetting that a ≠ 0 — a quadratic function must have a non-zero quadratic term.
忘记 a ≠ 0 — 二次函数必须含有非零的二次项。
Using the sign of b incorrectly when finding the axis of symmetry.
求对称轴时，b 的符号使用错误。
Confusing the direction of horizontal shift: f(x − h) shifts right, not left.
混淆水平平移的方向：f(x − h) 是向右平移，而非向左。
Misreading the vertex from a graph — always check the coordinates carefully.
从图像上读顶点坐标时看错 — 务必仔细核对坐标。
When completing the square, forgetting to adjust the constant term correctly.
配方时忘记正确调整常数项。
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