Examples of Graphs | 图表示例

📚 Examples of Graphs | 图表示例

Understanding the shapes and properties of different types of graphs is fundamental in IB Mathematics. From linear relationships to trigonometric oscillations, each graph tells a story about the functions they represent. This article walks you through essential examples of graphs you will encounter, along with their key features, transformations, and real-world connections.

理解不同类型图像的形状和性质是 IB 数学的基础。从线性关系到三角函数的振荡，每幅图像都在讲述其所代表的函数故事。本文将带你逐一了解你将遇到的重要图表示例，以及它们的关键特征、变换与现实联系。

---

1. Linear Graphs | 线性函数图像

A linear function has the form y = mx + c, where m is the gradient and c is the y-intercept. Its graph is a straight line. If m > 0, the line slopes upward; if m < 0, it slopes downward. Horizontal lines occur when m = 0 (y = c), and vertical lines are given by x = k (though not a function of x). The gradient m is constant, meaning the rate of change is uniform.

线性函数的形式为 y = mx + c，其中 m 是斜率，c 是 y 轴截距。它的图像是一条直线。当 m > 0 时，直线向上倾斜；当 m < 0 时，直线向下倾斜。当 m = 0 时得到水平线 y = c，竖直线由 x = k 给出（尽管不是 x 的函数）。斜率 m 是常数，意味着变化率是均匀的。

To sketch a line, find the y-intercept (0, c) and use the gradient to locate another point. For example, y = 2x – 3 crosses the y-axis at -3 and has a slope of 2, so moving 1 unit right and 2 units up gives a second point. The x-intercept occurs where y = 0, giving x = 3/2. Linear models are often used in economics and physics for simple relationships.

绘制直线时，先找到 y 轴截距 (0, c)，再利用斜率确定另一点。例如，y = 2x – 3 与 y 轴交于 -3，斜率为 2，向右移 1 单位、向上移 2 单位即可得到第二个点。x 轴截距在 y = 0 处，解得 x = 3/2。线性模型常用于经济学和物理学中的简单关系。

---

2. Quadratic Graphs | 二次函数图像

A quadratic function is of the form y = ax² + bx + c (a ≠ 0). Its graph is a parabola. If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point. The vertex is the turning point, and the axis of symmetry passes vertically through it. The y-intercept is c, and the x-intercepts are the roots of the equation ax² + bx + c = 0.

二次函数的形式为 y = ax² + bx + c（a ≠ 0）。它的图像是一条抛物线。若 a > 0，抛物线开口向上且有最小值点；若 a < 0，开口向下且有最大值点。顶点是转折点，对称轴垂直穿过顶点。y 轴截距为 c，x 轴截距是方程 ax² + bx + c = 0 的根。

The vertex can be found by completing the square: y = a(x – h)² + k, where (h, k) is the vertex. For example, y = 2x² – 4x + 1 becomes y = 2(x – 1)² – 1, so the vertex is (1, -1). The discriminant Δ = b² – 4ac tells us if the parabola cuts the x-axis: two distinct roots for Δ > 0, one repeated root for Δ = 0, no real roots for Δ < 0.

顶点可通过配方法求得：y = a(x – h)² + k，其中 (h, k) 即为顶点。例如，y = 2x² – 4x + 1 可化为 y = 2(x – 1)² – 1，顶点为 (1, -1)。判别式 Δ = b² – 4ac 表示抛物线与 x 轴的交点情况：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 无实根。

---

3. Cubic and Higher-Degree Polynomials | 三次及高次多项式图像

Cubic functions y = ax³ + bx² + cx + d can have up to three real roots and two turning points. The sign of a determines the end behaviour: if a > 0, the right-hand end goes to +∞; if a < 0, it goes to -∞. A typical cubic may possess a point of inflection instead of distinct turning points, as in y = x³. Higher-degree polynomials exhibit more complicated wiggles, but the number of turning points is at most degree – 1.

三次函数 y = ax³ + bx² + cx + d 最多有三个实根和两个转折点。a 的符号决定远端走势：若 a > 0，右端趋向 +∞；若 a < 0，趋向 -∞。典型的三次函数可能仅有一个拐点而非两个转折点，如 y = x³。更高次多项式会出现更复杂的起伏，但转折点个数最多为（次数 – 1）。

For example, y = x³ – 3x has roots at -√3, 0, √3 and turning points at (-1, 2) and (1, -2). A quartic such as y = x⁴ – 4x² has symmetry about the y-axis and up to three turning points. When sketching, identify zeros, end behaviour, and sign changes.

例如，y = x³ – 3x 的根在 -√3、0、√3，转折点位于 (-1, 2) 和 (1, -2)。四次函数如 y = x⁴ – 4x² 关于 y 轴对称，最多可有三个转折点。草绘时需识别零点、远端走势和符号变化。

---

4. Exponential Graphs | 指数函数图像

Exponential functions have the form y = aˣ (a > 0, a ≠ 1). If a > 1, the graph rises rapidly from near the x-axis; if 0 < a < 1, it decays towards zero. The x-axis is a horizontal asymptote (y = 0). The y-intercept is always (0, 1) because a⁰ = 1. For y = 2ˣ, the curve passes through (0, 1) and (1, 2), and grows without bound. For y = (1/2)ˣ, it is the reflection of y = 2ˣ across the y-axis.

指数函数的形式为 y = aˣ（a > 0，a ≠ 1）。若 a > 1，图像从 x 轴附近迅速上升；若 0 < a < 1，图像衰减趋向于零。x 轴是水平渐近线（y = 0）。y 轴截距始终为 (0, 1)，因为 a⁰ = 1。对于 y = 2ˣ，曲线经过 (0, 1) 和 (1, 2)，并无限增长。y = (1/2)ˣ 是 y = 2ˣ 关于 y 轴的反射。

The natural exponential y = eˣ, where e ≈ 2.718, is widely used in continuous growth models. The gradient at any point equals its value, which is unique. Exponential graphs never cross the horizontal asymptote and can model populations, radioactive decay, and compound interest.

自然指数函数 y = eˣ（e ≈ 2.718）广泛用于连续增长模型。任一点的斜率等于该点函数值，这是它独有的性质。指数图像永远不会穿过水平渐近线，可用于模拟种群、放射性衰变和复利。

---

5. Logarithmic Graphs | 对数函数图像

The logarithmic function y = logₐ x is the inverse of y = aˣ. Its domain is x > 0, and it has a vertical asymptote at x = 0. The graph passes through (1, 0) because logₐ 1 = 0. For a > 1, the curve increases slowly; for 0 < a < 1, it decreases. The y-axis acts as a vertical asymptote.

对数函数 y = logₐ x 是 y = aˣ 的反函数。其定义域为 x > 0，且在 x = 0 处有一条垂直渐近线。图像经过 (1, 0)，因为 logₐ 1 = 0。当 a > 1 时，曲线缓慢上升；当 0 < a < 1 时，曲线下降。y 轴是其垂直渐近线。

Natural logarithm y = ln x corresponds to base e. Its graph rises but becomes flatter as x increases. Logarithmic scales are used for data that spans many orders of magnitude, such as pH, sound intensity, and earthquake magnitudes. Solving equations like 2ˣ = 5 requires taking logs, often converting to linear form for graphical analysis.

自然对数 y = ln x 对应底数 e。其图像上升但随 x 增大而变得更为平缓。对数标度用于跨多个数量级的数据，如 pH 值、声强和地震烈度。求解方程如 2ˣ = 5 需要取对数，常常转换为线性形式以进行图像分析。

---

6. Trigonometric Graphs: Sine and Cosine | 正弦与余弦函数图像

The graphs of y = sin x and y = cos x are periodic with period 2π and amplitude 1. The sine graph starts at (0, 0), rises to a maximum of 1 at π/2, returns to 0 at π, hits -1 at 3π/2, and completes the cycle at 2π. The cosine graph starts at (0, 1) and has the same shape but shifted left by π/2. Both oscillate between -1 and 1.

y = sin x 和 y = cos x 的图像是周期为 2π、振幅为 1 的周期函数。正弦图像从 (0, 0) 开始，在 π/2 处达到最大值 1，在 π 处回到 0，在 3π/2 处达到 -1，并在 2π 完成一个周期。余弦图像从 (0, 1) 开始，形状相同，但向左平移了 π/2。两者的值均在 -1 和 1 之间振荡。

Transformations apply directly: y = A sin(Bx + C) + D has amplitude |A|, period 2π/|B|, phase shift -C/B, and vertical shift D. For example, y = 2 sin(3x – π) + 1 has amplitude 2, period 2π/3, phase shift π/3 to the right, and is shifted up by 1. These are crucial for modelling tides, sound waves, and seasonal data.

图像变换直接适用：y = A sin(Bx + C) + D 的振幅为 |A|，周期为 2π/|B|，相位平移为 -C/B，垂直平移为 D。例如，y = 2 sin(3x – π) + 1 的振幅为 2，周期为 2π/3，向右平移 π/3，向上平移 1。这些对于潮汐、声波和季节性数据建模至关重要。

---

7. Trigonometric Graphs: Tangent and Reciprocals | 正切及其他三角函数图像

The tangent function y = tan x = sin x / cos x has asymptotes where cos x = 0, at x = π/2 + kπ. Its period is π, and it passes through the origin with increasing steepness. Unlike sine and cosine, the tangent range is all real numbers. The graph repeats every π units and crosses the x-axis at multiples of π.

正切函数 y = tan x = sin x / cos x 在 cos x = 0 处，即 x = π/2 + kπ 处有渐近线。其周期为 π，通过原点且斜率不断增加。与正弦和余弦不同，正切的值域是所有实数。图像每 π 单位重复一次，且在 π 的整数倍处穿过 x 轴。

The reciprocal trigonometric functions—cosecant (csc x = 1/sin x), secant (sec x = 1/cos x), and cotangent (cot x = 1/tan x)—produce distinct graphs with vertical asymptotes where the denominator is zero. For instance, y = csc x has a U-shaped branch above y = 1 and an inverted branch below y = -1, excluding the strip between -1 and 1.

倒数三角函数——余割（csc x = 1/sin x）、正割（sec x = 1/cos x）和余切（cot x = 1/tan x）产生独特的图像，在分母为零处有垂直渐近线。例如，y = csc x 在 y = 1 上方有 U 形分支，在 y = -1 下方有倒 U 形分支，排除了 -1 到 1 之间的带状区域。

---

8. Absolute Value Graphs | 绝对值函数图像

The absolute value function y = |x| is V-shaped with vertex at (0, 0). It reflects the negative part of y = x across the x-axis to make all y-values non-negative. More generally, y = |f(x)| takes any negative portion of f(x) and reflects it above the x-axis. The parts where f(x) ≥ 0 remain unchanged.

绝对值函数 y = |x| 呈 V 形，顶点在 (0, 0)。它将 y = x 的负值部分关于 x 轴反射，使得所有 y 值非负。更一般地，y = |f(x)| 将 f(x) 的任意负值部分向上反射至 x 轴上方。f(x) ≥ 0 的部分保持不变。

For example, y = |x – 2| shifts the V-shape right by 2 units, while y = |x² – 4| takes the parabola y = x² – 4, which goes below the x-axis between -2 and 2, and reflects that segment upward, creating a W-like shape. Absolute value graphs are helpful for modelling distances and magnitude errors.

例如，y = |x – 2| 将 V 形向右平移 2 个单位，而 y = |x² – 4| 将抛物线 y = x² – 4 在 -2 到 2 之间位于 x 轴下方的部分向上反射，形成类似 W 的形状。绝对值图像对于距离和误差幅度建模十分有用。

---

9. Rational Function Graphs | 有理函数图像

Rational functions are quotients of polynomials, e.g., y = 1/x. The graph of y = 1/x is a hyperbola with asymptotes x = 0 and y = 0. It lies in the first and third quadrants for y = 1/x. For y = 1/x², the graph is in the first and second quadrants, symmetric about the y-axis. These graphs demonstrate asymptotic behaviour and discontinuities.

有理函数是多项式的商，例如 y = 1/x。y = 1/x 的图像是双曲线，具有渐近线 x = 0 和 y = 0。y = 1/x 的图像位于第一和第三象限。对于 y = 1/x²，图像位于第一和第二象限，关于 y 轴对称。这些图像展示了渐近行为和不连续性。

More complex rational functions like y = (x + 1)/(x – 2) have a vertical asymptote at x = 2 and a horizontal asymptote at y = 1 (since degrees are equal, ratio of leading coefficients). Finding intercepts is key: x-intercept when numerator is zero, y-intercept when x = 0. Sign diagrams help sketch the branches correctly.

更复杂的有理函数如 y = (x + 1)/(x – 2) 在 x = 2 处有垂直渐近线，在 y = 1 处有水平渐近线（因为分子分母次数相同，取首项系数之比）。求截距是关键：分子为零得 x 截距，x = 0 得 y 截距。符号表有助于正确绘制各分支。

---

10. Transformations of Graphs | 图像的变换

Transformations allow us to sketch related graphs quickly. For a function y = f(x), the transformation y = f(x) + k shifts the graph vertically by k units. y = f(x + h) shifts horizontally by -h units (left if h > 0, right if h < 0). y = -f(x) reflects across the x-axis, while y = f(-x) reflects across the y-axis.

图像变换使我们能快速绘制相关函数的图形。对于函数 y = f(x)，变换 y = f(x) + k 将图像垂直平移 k 个单位。y = f(x + h) 水平平移 -h 个单位（h > 0 向左，h < 0 向右）。y = -f(x) 关于 x 轴对称反射，y = f(-x) 关于 y 轴对称反射。

Stretches are given by y = af(x) (vertical stretch by factor a) and y = f(bx) (horizontal stretch by factor 1/b). Combining transformations follows the order: horizontal shifts, stretches, reflections, then vertical shifts. For example, y = 2 f(x – 3) stretches vertically by 2 and shifts right by 3.

伸缩变换由 y = af(x)（垂直伸缩 a 倍）和 y = f(bx)（水平伸缩 1/b 倍）给出。组合变换遵循顺序：水平平移、伸缩、反射，最后垂直平移。例如，y = 2 f(x – 3) 将图像垂直拉伸 2 倍，并向右平移 3 个单位。

In IB, you often apply these to trigonometric, exponential, and polynomial graphs. Understanding the mapping of key points (e.g., vertices, asymptotes) under transformations is a powerful sketch tool.

在 IB 中，你通常需要将这些变换应用于三角函数、指数函数和多项式图像。理解关键点（如顶点、渐近线）在变换下的映射是一个强大的绘图工具。

---

11. Graphs of Inverse Functions | 反函数的图像

If a function f is invertible, the graph of its inverse f⁻¹ is the reflection of y = f(x) across the line y = x. This means every point (a, b) on f maps to (b, a) on f⁻¹. For the inverse to exist as a function, f must be one-to-one.

若函数 f 可逆，其反函数 f⁻¹ 的图像是 y = f(x) 关于直线 y = x 的反射。这意味着 f 上的每一点 (a, b) 映射到 f⁻¹ 上的 (b, a)。要使反函数作为函数存在，f 必须是一对一的。

Common examples: y = eˣ and y = ln x are inverses; their graphs reflect over y = x. The domain of f becomes the range of f⁻¹, and vice versa. When sketching, draw the mirror line y = x as a guide. You can also restrict the domain of a non-injective function (like y = x² for x ≥ 0) to make it invertible and then reflect the relevant portion.

常见示例：y = eˣ 与 y = ln x 互为反函数；它们的图像关于 y = x 对称。f 的定义域成为 f⁻¹ 的值域，反之亦然。草图时可先绘制镜像线 y = x 作为引导。你也可以限制非单射函数的定义域（如将 y = x² 限制在 x ≥ 0），使其可逆，然后反射相应部分。

---

12. Using Graphs to Solve Equations | 利用图像解方程

Graphs provide a visual method for solving equations. The solutions to f(x) = 0 are the x-intercepts of y = f(x). To solve f(x) = g(x), plot both functions and find their intersection points; the x-coordinates are the solutions. This is especially useful when algebraic methods are challenging.

图像为解方程提供了一种直观方法。方程 f(x) = 0 的解就是 y = f(x) 的 x 轴截距。要解 f(x) = g(x)，画出两个函数的图像并找到它们的交点；交点的 x 坐标即为解。这在代数方法困难时尤其有用。

IB exam questions may ask you to sketch a graph and hence estimate roots or intersections, or to identify the number of solutions. For example, the equation eˣ = 3 – x can be solved by intersecting y = eˣ and y = 3 – x. The graphical approach also reinforces understanding of function behaviour and inequalities.

IB 考题可能会要求你绘制图像并据此估计根或交点的个数。例如，方程 eˣ = 3 – x 可以借助 y = eˣ 和 y = 3 – x 的交点来求解。图像方法还能加深对函数性质和不等式的理解。

Pay attention to asymptotes and end behaviour, as they can indicate when an equation has no real solution. Always label your axes and key points clearly in exam sketches.

注意渐近线和远端走势，它们可以提示方程是否无实数解。在考试绘图中务必清晰标注坐标轴和关键点。

---

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）