📚 Quadratic Graphs | 二次函数图像
A quadratic graph is the visual representation of a quadratic function, one of the most fundamental curves in A-Level Mathematics. Understanding its shape, key points, and transformations is essential for topics ranging from algebraic manipulation to calculus and modelling.
二次函数图像是二次函数的视觉表达，也是 A-Level 数学中最基本的曲线之一。理解其形状、关键点和变换，对于从代数运算到微积分和数学建模的诸多主题都至关重要。
1. Introduction to Quadratic Functions | 二次函数简介
ith a, b, c real constants and a ≠ 0. The graph of such a function is a smooth, symmetric U‑shaped curve called a parabola.
二次函数是二次多项式，最常写成一般式 f(x) = ax² + bx + c，其中 a, b, c 为实数常数且 a ≠ 0。该函数的图像是一条光滑、对称的 U 形曲线，称为抛物线。
Every parabola has a vertex (the turning point), an axis of symmetry that passes through the vertex, and may intersect the x‑axis at zero, one, or two points depending on the nature of its roots.
每条抛物线都有一个顶点（转折点）、一条穿过顶点的对称轴，并且根据根的情况，可能与 x 轴有零个、一个或两个交点。
2. Standard Form and Shape | 标准形式和形状
the sign of the leading coefficient a dictates the opening direction. If a > 0, the parabola opens upwards like a regular 'U'; if a < 0, it opens downwards like an upside‑down 'U'.
在一般式 f(x) = ax² + bx + c 中，首项系数 a 的符号决定了开口方向。若 a > 0，抛物线开口向上，呈正 U 形；若 a < 0，则开口向下，呈倒 U 形。
The magnitude |a| also controls the 'width' of the parabola. Larger values of |a| make the graph steeper and narrower, while smaller values of |a| (between 0 and 1) produce a wider, shallower curve.
系数 |a| 的大小还控制抛物线的“宽度”。|a| 越大，图像越陡、越窄；|a| 越小（0 到 1 之间），曲线越宽、越平缓。
3. Vertex and Axis of Symmetry | 顶点和对称轴
owest point of the parabola and lies on the axis of symmetry. For the standard form, the x‑coordinate of the vertex is given by x = -b / (2a). The y‑coordinate is obtained by substituting this x back into the function.
顶点是抛物线的最高点或最低点，位于对称轴上。对于一般式，顶点的 x 坐标由 x = -b / (2a) 给出；y 坐标则将此 x 代入函数求得。
For example, for f(x) = 2x² + 4x - 6, we have a = 2, b = 4. The x‑ 6 = -8. Hence the vertex is (-1, -8) and the axis of symmetry is the vertical line x = -1.
例如，对于 f(x) = 2x² + 4x - 6，a = 2, b = 4。顶点的 x 坐标为 x = -4 / (2 × 2) = -1。代入得 y = 2(-1)² + 4(-1) - 6 = -8。因此顶点为 (-1, -8)，对称轴为竖直线 x = -1。
of symmetry x = h. Converting between standard and vertex forms is a key algebraic skill.
二次函数的顶点式 y = a(x - h)² + k 可直接读出顶点 (h, k) 及对称轴 x = h。在一般式与顶点式之间进行转换是关键的代数技能。
4. Finding Roots: The Discriminant | 求根：判别式
he solutions to ax² + bx + c = 0. They can be found using the quadratic formula: x = (-b ± √b² - 4ac) / (2a). The expression under the square root, Δ = b² - 4ac, is called the discriminant.
二次方程的根（或零点）是 ax² + bx + c = 0 的解。可通过求根公式求出：x = (-b ± √b² - 4ac) / (2a)。根号内的表达式 Δ = b² - 4ac 称为判别式。
The discriminant determines the nature of the roots and the number of x‑axis intersections:
判别式决定了根的性质以及与 x 轴交点的个数：
- If Δ > 0, there are two distinct real roots; the parabola cuts the x‑axis at two points.
- 如果 Δ > 0，有两个不相等的实根；抛物线与 x 轴有两个交点。
- If Δ = 0, there is one repeated real root; the parabola touches (is tangent to) the x‑axis at the vertex.
- 如果 Δ = 0，有一个重根；抛物线在顶点处与 x 轴相切。
- If Δ < 0, there are no real roots; the parabola lies entirely above or entirely below the x‑axis.
- 如果 Δ < 0，没有实根；抛物线完全位于 x 轴上方或下方。
5 ± 1)/2, i.e. x = 3 and x = 2. In contrast, x² + x + 1 has Δ = 1 - 4 = -3 < 0, hence no x‑intercepts.
例如，f(x) = x² - 5x + 6 的 Δ = 25 - 24 = 1 > 0，故根为 x = (5 ± 1)/2，即 x = 3 和 x = 2。而 x² + x + 1 的 Δ = -3 < 0，因此没有 x 轴截距。
5. Intercepts: x‑intercepts and y‑intercept | 截距：x 轴与 y 轴截距
The y‑ x) = ax² + bx + c, giving the point (0, c). This is always a single point on the parabola.
y 轴截距可直接令 x = 0 代入 f(x) = ax² + bx + c 得到，为点 (0, c)。这总是抛物线上的一个点。
The x‑intercepts, if they exist, are the real roots of the quadratic. Graphically, they are the points (α, 0) and (β, 0) where the parabola meets the x‑axis. Their presence and number are governed by the discriminant.
x 轴截距（若存在）是二次方程的实根。从图像上看，它们是抛物线与 x 轴相交的点 (α, 0) 和 (β, 0)。其有无与个数由判别式决定。
Example: f(x) = x² - 4x + 3 has y‑intercept (0, 3) and x‑ 1)(x - 3) = 0, so x = 1 and x = 3. The intercepts help in sketching the graph accurately.
举例：f(x) = x² - 4x + 3 的 y 轴截距为 (0, 3)，解 x² - 4x + 3 = 0 得 x 截距为 x = 1 和 x = 3。这些截距有助于精确绘制图像。
6. Sketching Quadratic Graphs | 绘制二次函数图像
To sketch a quadratic graph without a calculator, follow a systematic approach:
要徒手绘制二次函数图像，可遵循以下系统步骤：
Step 1: Identify the shape by checking the sign of a (upwards for a > 0, downwards for a < 0).
第 1 步：通过 a 的符号判断形状（a > 0 开口向上，a < 0 开口向下）。
Step 2: Find the y‑intercept (0, c).
第 2 步：求 y 轴截距 (0, c)。
Step 3: Calculate the vertex using x = -b/(2a) and compute its y‑coordinate.
第 3 步：用 x = -b/(2a) 计算顶点 x 坐标并求出 y 坐标。
Step 4: Determine x‑intercepts by solving ax² + bx + c = 0, if real roots exist.
第 4 步：若存在实根，解 ax² + bx + c = 0 求 x 轴截距。
Step 5: Draw the axis of symmetry as a dashed vertical line through the vertex.
第 5 步：画出穿过顶点的对称轴（竖虚线）。
Step 6: Plot the vertex, intercepts, and a few symmetric points. Then draw a smooth curve.
第 6 步：标出顶点、截距及一些对称点，然后描出光滑曲线。
For example, sketch f(x) = -x² + 2x + 3. a = -1 (opens down), y‑ ves x = -1 and x = 3. Plot these points and draw a downward parabola passing through them.
例如，绘制 f(x) = -x² + 2x + 3。a = -1（开口向下），y 轴截距 (0, 3)。顶点：x = -2/(2 × -1) = 1，y = 4，得 (1, 4)。解方程得 x 轴截距 x = -1 和 x = 3。标出这些点，画出经过它们的向下抛物线。
7. Completing the Square and Vertex Form | 配方法和顶点式
- h)² + k, immediately showing the vertex (h, k) and making it easier to analyse the function.
配方法可将 f(x) = ax² + bx + c 转化为顶点式 a(x - h)² + k，直接显示顶点 (h, k)，便于分析函数。
For example, x² + 6x + 5: b/2 = 3, so (x + 3)² - 9 + 5 = (x + 3)² - 4. Vertex is (-3, -4).
当 a = 1 时，对 x² + bx 加上并减去 (b/2)²：x² + bx = (x + b/2)² - (b/2)²，再加上 c。例如 x² + 6x + 5：b/2 = 3，得 (x + 3)² - 9 + 5 = (x + 3)² - 4，顶点为 (-3, -4)。
2)² - 1. Vertex is (-2, -1). This method is essential for finding maximum/minimum values.
若 a ≠ 1，先从前两项提公因子 a：如 2x² + 8x + 7 = 2(x² + 4x) + 7。对内配方：2[(x + 2)² - 4] + 7 = 2(x + 2)² - 1，顶点为 (-2, -1)。该方法是求最大值/最小值的基础。
8. Maximum and Minimum Values | 最大值和最小值
The vertex represents either the maximum or the minimum of the quadratic function, depending on the sign of a. For a > 0, the parabola opens upwards and the vertex gives the global minimum value, which is the y‑coordinate of the vertex. For a < 0, it opens downwards and the vertex gives the global maximum.
顶点代表二次函数的最大值或最小值，取决于 a 的符号。对于 a > 0，抛物线开口向上，顶点给出全局最小值，即顶点的 y 坐标。对于 a < 0，抛物线开口向下，顶点给出全局最大值。
² + 2 has a minimum of 2 when x = 5. f(x) = -2(x + 1)² + 7 has a maximum of 7 when x = -1.
在顶点式 y = a(x - h)² + k 中，最大值或最小值即为 k，在 x = h 处取得。例如，f(x) = 3(x - 5)² + 2 在 x = 5 时有最小值 2。f(x) = -2(x + 1)² + 7 在 x = -1 时有最大值 7。
These extreme values are extremely useful in optimisation problems, such as maximising area or minimising cost, where a real‑world situation is modelled by a quadratic.
这些极值在优化问题中极为有用，例如用二次函数对实际情境建模以最大化面积或最小化成本。
9. Quadratic Inequalities and Graphs | 二次不等式与图像
ficiently using the graph of the corresponding quadratic. By finding the roots and knowing the shape of the parabola, the sign of the function over different intervals becomes obvious.
二次不等式如 ax² + bx + c > 0 或 ax² + bx + c ≤ 0 可借助相应二次函数的图像高效求解。求出根并了解抛物线形状后，函数在不同区间的正负便一目了然。
. The roots are x = 1 and x = 3, and a = 1 > 0 (opens upward). The parabola is below the x‑ ween the roots and above it outside. Hence the inequality is satisfied for x < 1 or x > 3.
考虑 x² - 4x + 3 > 0。根为 x = 1 和 x = 3，a = 1 > 0（开口向上）。抛物线在两根之间位于 x 轴下方，之外位于上方。因此不等式解为 x < 1 或 x > 3。
quick graph to avoid sign errors, especially when multiplying or dividing by negative numbers in algebraic approaches.
对于 x² - 4x + 3 ≤ 0，解为闭区间 1 ≤ x ≤ 3。应始终快速画出示意图，以避免符号错误，尤其在用代数方法乘除负数时。
10. Transformations of Quadratic Graphs | 二次函数图像的变换
nderstanding these transformations helps in sketching and interpreting modified functions.
任何二次函数的图像都可视为基本抛物线 y = x² 的变换。理解这些变换有助于绘制和解读修改后的函数。
f k > 0, down if k < 0) from y = ax². The factor a stretches or compresses the graph vertically: |a| > 1 makes it narrower, 0 < |a| < 1 makes it wider. A negative a reflects the graph in the x‑axis.
y = a(x - h)² + k 形式的函数涉及从 y = ax² 出发的水平平移 h 个单位（h > 0 向右，h < 0 向左）和竖直平移 k 个单位（k > 0 向上，k < 0 向下）。系数 a 在竖直方向上拉伸或压缩图像：|a| > 1 使图像变窄，0 < |a| < 1 使图像变宽。a 为负则使图像关于 x 轴反射。
ts a reflection (negative), a vertical compression by factor ½, a shift left by 3, and a shift down by 2. The vertex moves from (0,0) to (-3, -2), and the shape is a wide upside‑down parabola.
例如，y = -½(x + 3)² - 2 表示一次反射（负号）、竖直方向缩为 ½、向左平移 3、向下平移 2。顶点从 (0,0) 移至 (-3, -2)，形状为宽的倒抛物线。
By mastering these transformations, you can now visualise and sketch any quadratic graph with ease, connecting algebraic expressions to their geometric meaning
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