📚 Quadratic Function Graphs: Direction, Vertex, and Axis of Symmetry | 二次函数图像：开口、顶点与对称轴
Quadratic functions represent one of the most fundamental topics in algebra. Their graphs, called parabolas, appear throughout mathematics and physics. Understanding how the coefficients in the standard form affect the shape, direction, and position of the parabola is essential for solving problems involving maximum and minimum values, motion under gravity, and optimisation.
二次函数是代数中最基础的内容之一，其图像被称为抛物线，在数学与物理中广泛出现。理解标准形式中的系数如何影响抛物线的形状、开口方向与位置，是解决最大值与最小值问题、重力运动以及优化问题的基础。
1. The Standard and Vertex Forms | 标准形式与顶点式
The general form of a quadratic function is f(x) = ax² + bx + c, where a, b and c are real constants and a ≠ 0. The graph of this function is always a parabola. The coefficient a determines the direction of the opening, while b and c affect the position of the graph in the coordinate plane.
二次函数的一般形式为 f(x) = ax² + bx + c，其中 a、b、c 为实数常数，且 a ≠ 0。该函数的图像始终是一条抛物线。系数 a 决定开口方向，b 和 c 影响图像在坐标平面中的位置。
Another important form is the vertex form: f(x) = a(x − h)² + k. In this form, the coordinates of the vertex are immediately visible: the vertex is at the point (h, k). This form is particularly useful for graphing because it directly reveals the turning point of the parabola.
另一种重要形式为顶点式：f(x) = a(x − h)² + k。在此形式中，顶点坐标一目了然：顶点位于点 (h, k)。该形式便于直接绘制图像，因为它直接给出了抛物线的转折点。
To convert from the standard form to the vertex form, the technique of completing the square is used. For example, f(x) = x² − 6x + 5 can be rewritten as f(x) = (x − 3)² − 4, revealing that the vertex is at (3, −4).
将标准形式转化为顶点式需要使用配方法。例如，f(x) = x² − 6x + 5 可改写为 f(x) = (x − 3)² − 4，从而得出顶点为 (3, −4)。
2. Direction of Opening | 开口方向
The sign of the leading coefficient a determines which way the parabola opens. If a > 0, the parabola opens upward, creating a shape like a valley. This means the vertex is the minimum point of the function. If a < 0, the parabola opens downward, creating a shape like a mountain, and the vertex is the maximum point.
首项系数 a 的符号决定抛物线的开口方向。若 a > 0，抛物线开口向上，形如山谷，此时顶点为函数的最小值点。若 a < 0，抛物线开口向下，形如山丘，此时顶点为函数的最大值点。
Consider the functions f(x) = x² and g(x) = −x². The first opens upward with a minimum at the origin, while the second opens downward with a maximum at the origin. Their graphs are mirror reflections of each other across the x-axis.
考虑函数 f(x) = x² 和 g(x) = −x²。前者开口向上，在原点处取得最小值；后者开口向下，在原点处取得最大值。二者的图像关于 x 轴对称，互为镜像。
a > 0 ⟹ opens upward (minimum at vertex)
a < 0 ⟹ opens downward (maximum at vertex)
a > 0 ⟹ 开口向上（顶点为最小值）
a < 0 ⟹ 开口向下（顶点为最大值）
3. Width of the Parabola | 抛物线的宽窄
The absolute value of a, written as |a|, controls how wide or narrow the parabola appears. When |a| is large, the parabola is narrow and steep. When |a| is small, the parabola is wide and flat. This happens because larger values of a cause the y-values to grow more quickly as x moves away from the vertex.
|a|（即 a 的绝对值）控制抛物线的宽窄。当 |a| 较大时，抛物线狭窄而陡峭；当 |a| 较小时，抛物线宽阔而平缓。这是因为较大的 a 值会使 y 值随着 x 远离顶点而更快地增长。
For instance, the graph of f(x) = 3x² is much narrower than the graph of f(x) = ½x². Both have the same vertex at the origin and both open upward, yet the points on the first parabola rise three times faster than those on the second.
例如，f(x) = 3x² 的图像比 f(x) = ½x² 的图像狭窄得多。两者顶点均为原点且都开口向上，但第一条抛物线上的点上升速度是第二条的三倍。
It should be noted that the width of the parabola does not depend on b or c. Only a determines the vertical scaling of the graph.
需要注意的是，抛物线的宽窄不依赖于 b 或 c，只有 a 决定图像的纵向伸缩比例。
4. The Vertex and the Formula x = −b/(2a) | 顶点与公式 x = −b/(2a)
The vertex of a parabola is its turning point, where the function changes from increasing to decreasing or vice versa. The x-coordinate of the vertex can be found using the formula x = −b/(2a). Once the x-coordinate is known, the y-coordinate is obtained by substituting this value back into the original function.
抛物线的顶点是它的转折点，函数在此处从递增变为递减或从递减变为递增。顶点的 x 坐标可通过公式 x = −b/(2a) 求得。得到 x 坐标后，将其代回原函数即可求得 y 坐标。
For example, given f(x) = 2x² − 8x + 3, the x-coordinate of the vertex is x = −(−8)/(2×2) = 2. Substituting x = 2 gives f(2) = 2(4) − 16 + 3 = −5. Therefore, the vertex is at (2, −5).
例如，对于 f(x) = 2x² − 8x + 3，顶点的 x 坐标为 x = −(−8)/(2×2) = 2。将 x = 2 代入得 f(2) = 2(4) − 16 + 3 = −5。因此，顶点为 (2, −5)。
This formula comes from completing the square on the general form ax² + bx + c. The result shows that the vertex is always located at the point where the derivative of the function is zero.
这一公式来源于对一般形式 ax² + bx + c 进行配方。结果表明，顶点始终位于函数导数为零的位置。
5. Axis of Symmetry | 对称轴
Every parabola is symmetric about a vertical line called the axis of symmetry. This line passes through the vertex and has the equation x = −b/(2a). The two halves of the parabola on either side of this line are mirror images of each other.
每条抛物线都关于一条垂直直线对称，这条直线称为对称轴。对称轴经过顶点，其方程为 x = −b/(2a)。位于该直线两侧的两半抛物线互为镜像。
If a quadratic function has two real roots r₁ and r₂, the axis of symmetry is also the average of the roots: x = (r₁ + r₂)/2. This connection between the roots and the axis of symmetry is particularly useful when the roots are easy to identify.
若二次函数有两个实数根 r₁ 和 r₂，则对称轴也等于两根的平均值：x = (r₁ + r₂)/2。根与对称轴之间的这种联系在根易于求出时特别有用。
For the function f(x) = x² − 4x + 3, the roots are x = 1 and x = 3, so the axis of symmetry is x = (1+3)/2 = 2, which matches the result from the formula −b/(2a).
对于函数 f(x) = x² − 4x + 3，根为 x = 1 和 x = 3，因此对称轴为 x = (1+3)/2 = 2，与公式 −b/(2a) 的结果一致。
6. Sketching the Graph | 绘制图像
To sketch the graph of a quadratic function, the following steps are recommended. First, determine the direction of opening by checking the sign of a. Second, find the axis of symmetry and the vertex. Third, find the y-intercept by evaluating f(0). Fourth, if possible, find the x-intercepts by solving f(x) = 0. Finally, plot the vertex, intercepts, and one or two additional points, then draw a smooth curve through them.
绘制二次函数图像时，建议按以下步骤进行。首先，通过 a 的符号判断开口方向。其次，求对称轴与顶点。第三，计算 f(0) 得 y 截距。第四，若可能，解方程 f(x) = 0 得 x 截距。最后，标出顶点、截距及一两个附加点，并通过它们画一条平滑曲线。
For example, to sketch f(x) = −x² + 2x + 3, note that a = −1, so the parabola opens downward. The axis of symmetry is x = −2/(2×(−1)) = 1, and the vertex is at (1, 4). The y-intercept is 3, and the x-intercepts are found by solving −x² + 2x + 3 = 0, which gives x = 3 and x = −1.
例如，绘制 f(x) = −x² + 2x + 3 时，a = −1，因此抛物线开口向下。对称轴为 x = −2/(2×(−1)) = 1，顶点为 (1, 4)。y 截距为 3，解方程 −x² + 2x + 3 = 0 得 x 截距为 x = 3 和 x = −1。
Key features: direction (a), axis (x = −b/2a), vertex, y-intercept, x-intercepts
关键要素：开口方向 (a)、对称轴 (x = −b/2a)、顶点、y 截距、x 截距
7. Relationship with the Discriminant | 与判别式的关系
The discriminant, denoted Δ and defined as Δ = b² − 4ac, determines how many x-intercepts the parabola has. If Δ > 0, there are two distinct real roots, so the parabola crosses the x-axis at two points. If Δ = 0, there is exactly one real root, so the parabola touches the x-axis at exactly one point — the vertex. If Δ < 0, there are no real roots, so the parabola never touches the x-axis.
判别式 Δ 定义为 Δ = b² − 4ac，它决定抛物线与 x 轴的交点数量。若 Δ > 0，有两个不相等的实数根，抛物线在两点处穿过 x 轴。若 Δ = 0，恰好有一个实数根，抛物线在顶点处与 x 轴相切。若 Δ < 0，没有实数根，抛物线不与 x 轴相交。
The discriminant does not affect the direction or the width of the parabola. Two parabolas with the same a and b but different c values have the same axis of symmetry but potentially different numbers of x-intercepts.
判别式不影响抛物线的开口方向与宽窄。具有相同 a 和 b 但不同 c 值的两条抛物线拥有相同的对称轴，但 x 截距的数量可能不同。
8. Maximum and Minimum Values | 最大值与最小值
At the vertex, the quadratic function attains either its maximum or minimum value. If a > 0, the vertex represents the minimum value of the function, given by k = f(−b/(2a)). If a < 0, the vertex represents the maximum value. These extrema are crucial in optimisation problems where a quantity is to be maximised or minimised.
在顶点处，二次函数取得最大值或最小值。若 a > 0，顶点代表函数的最小值，其值为 k = f(−b/(2a))。若 a < 0，顶点代表最大值。这些极值在处理最大化或最小化问题的优化场景中至关重要。
For example, the height of a projectile thrown upward is modelled by h(t) = −5t² + 20t + 2 metres. Its maximum height occurs at t = −20/(2×(−5)) = 2 seconds, and h(2) = −20 + 40 + 2 = 22 metres.
例如，一个向上抛出的物体其高度由 h(t) = −5t² + 20t + 2 米建模。最大高度发生在 t = −20/(2×(−5)) = 2 秒时，且 h(2) = −20 + 40 + 2 = 22 米。
a > 0: minimum at vertex = c − b²/(4a)
a < 0: maximum at vertex = c − b²/(4a)
a > 0：顶点处取最小值 = c − b²/(4a)
a < 0：顶点处取最大值 = c − b²/(4a)
9. Transformations of the Parabola | 抛物线的变换
The graph of y = a(x − h)² + k is obtained from the basic graph y = x² through a series of transformations. The parameter a stretches or compresses the graph vertically and reflects it across the x-axis if negative. The parameter h shifts the graph horizontally: to the right if h > 0, to the left if h < 0. The parameter k shifts the graph vertically: upward if k > 0, downward if k < 0.
y = a(x − h)² + k 的图像可由基本图像 y = x² 经过一系列变换得到。参数 a 对图像进行纵向拉伸或压缩，若为负数则同时关于 x 轴反射。参数 h 对图像进行水平平移：h > 0 时向右移，h < 0 时向左移。参数 k 对图像进行垂直平移：k > 0 时向上移，k < 0 时向下移。
It is important to note the sign in the vertex form. In y = a(x + 3)² − 2, the horizontal shift is 3 units to the left, not to the right, because the expression is x − (−3). The vertex is located at (−3, −2).
需要注意顶点式中的正负号。在 y = a(x + 3)² − 2 中，水平平移为向左 3 个单位，而非向右，因为表达式为 x − (−3)。顶点位于 (−3, −2)。
10. Solving Problems from the Graph | 从图像解题
Given the graph of a quadratic function, one can read off the vertex, the axis of symmetry, the intercepts, and the direction of opening. These features allow us to answer questions about the range of the function, the intervals where the function is increasing or decreasing, and the conditions under which f(x) > 0 or f(x) < 0.
给定二次函数的图像后，我们可以直接读出顶点、对称轴、截距与开口方向。这些特征使我们能够回答关于函数值域、函数递增或递减区间，以及 f(x) > 0 或 f(x) < 0 的条件等问题。
If the parabola opens upward and the vertex is at (2, −5), then the range is [−5, ∞). The function is decreasing on (−∞, 2) and increasing on (2, ∞). Conversely, if the parabola opens downward, the range is (−∞, k] and the monotonic intervals are reversed.
若抛物线开口向上且顶点为 (2, −5)，则值域为 [−5, ∞)。函数在 (−∞, 2) 上递减，在 (2, ∞) 上递增。反之，若抛物线开口向下，则值域为 (−∞, k]，单调区间正好相反。
To determine where f(x) > 0, locate these intervals on the x-axis where the graph lies above the x-axis. If the parabola opens upward and has roots at x = −1 and x = 3, then f(x) > 0 for x < −1 or x > 3.
要判断 f(x) > 0 的范围，找出图像位于 x 轴上方的区间即可。若抛物线开口向上且根为 x = −1 与 x = 3，则当 x < −1 或 x > 3 时 f(x) > 0。
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