Methods for Finding Intersections of Function Graphs | 求函数图像交点的方法

📚 Methods for Finding Intersections of Function Graphs | 求函数图像交点的方法

When two functions are graphed on the same coordinate plane, their intersection points are the coordinates that satisfy both equations at the same time. In IB Mathematics, finding these points is a fundamental skill that connects algebra, graphing and problem solving.

当两个函数画在同一坐标系中时，它们的交点就是同时满足两个方程的点坐标。在 IB 数学中，求交点是连接代数、图像以及应用题的重要基础技能。

1. What Is an Intersection Point? | 交点的意义

Suppose we have two functions y = f(x) and y = g(x). A point (a, b) lies on the graph of f if b = f(a), and it lies on the graph of g if b = g(a). Therefore, an intersection point exists when f(a) = g(a) and b equals that common value.

设有两个函数 y = f(x) 和 y = g(x)。若点 (a, b) 在 f 的图像上，则 b = f(a)；若它也在 g 的图像上，则 b = g(a)。因此，当 f(a) = g(a) 时，该点同时位于两个图像上，这个公共点就是交点。

The x-coordinate of every intersection point must be a solution of the equation f(x) = g(x). Once this x-value is found, substituting it into either function gives the corresponding y-coordinate.

交点的 x 坐标一定是方程 f(x) = g(x) 的解。求出 x 值后，再把它代入任意一个函数中，就能得到对应的 y 坐标。

2. The Core Method: Set the Functions Equal | 核心方法：令两个函数相等

To find where y = f(x) and y = g(x) intersect, start by solving f(x) = g(x).

要求 y = f(x) 与 y = g(x) 的交点，首先解方程 f(x) = g(x)。

f(x) = g(x)

Use these steps:

按以下步骤操作：

Write both functions in the form y = expression in x.

把两个函数都写成 y = 含 x 的表达式形式。
Set the two expressions equal to each other.

令两个表达式相等。
Solve the resulting equation for x.

解所得方程，求出 x。
Substitute each x-value into either original function to find y.

把每个 x 值代入原函数中的一个，求出 y。
Write the final answer as coordinate pair(s).

将答案写成坐标形式。

This method works for linear, quadratic, rational, exponential and trigonometric functions, as long as the resulting equation can be solved.

这个方法对一次函数、二次函数、分式函数、指数函数和三角函数都适用，只要最后得到的方程能够求解。

3. Linear-Linear Intersections | 一次函数与一次函数的交点

For two linear functions y = m₁x + c₁ and y = m₂x + c₂, set the expressions equal:

对于两个一次函数 y = m₁x + c₁ 和 y = m₂x + c₂，令两个表达式相等：

m₁x + c₁ = m₂x + c₂

Rearrange to isolate x. If m₁ ≠ m₂, there is exactly one intersection point. If m₁ = m₂ and c₁ ≠ c₂, the lines are parallel and there is no intersection. If m₁ = m₂ and c₁ = c₂, the lines are identical and there are infinitely many common points.

移项并解出 x。若 m₁ ≠ m₂，则只有一个交点；若 m₁ = m₂ 且 c₁ ≠ c₂，则两直线平行，没有交点；若 m₁ = m₂ 且 c₁ = c₂，则两条直线重合，有无数个公共点。

Example: Find the intersection of y = 2x + 1 and y = −x + 4.

例：求 y = 2x + 1 与 y = −x + 4 的交点。

2x + 1 = −x + 4

This gives 3x = 3, so x = 1. Then y = 2(1) + 1 = 3. The intersection point is (1, 3).

由此得到 3x = 3，所以 x = 1。于是 y = 2(1) + 1 = 3。交点坐标为 (1, 3)。

4. Quadratic-Linear Intersections and the Discriminant | 二次函数与一次函数的交点及判别式

When a quadratic function y = ax² + bx + c intersects a linear function y = mx + k, substitute the linear expression into the quadratic equation:

当二次函数 y = ax² + bx + c 与一次函数 y = mx + k 相交时，把一次式代入二次方程：

ax² + bx + c = mx + k

After rearranging, you obtain a quadratic equation of the form Ax² + Bx + C = 0. The number of real intersections can be determined by the discriminant Δ = B² − 4AC.

整理后得到形如 Ax² + Bx + C = 0 的二次方程。实数交点的个数可由判别式 Δ = B² − 4AC 判断。

Discriminant Δ	Number of Intersections	Geometric Meaning
Δ > 0	Two distinct points	The line cuts the curve twice
Δ = 0	One repeated point	The line is tangent to the curve
Δ < 0	No real point	The line does not meet the curve

Example: Find the intersections of y = x² − 3x + 2 and y = x − 1.

例：求 y = x² − 3x + 2 与 y = x − 1 的交点。

x² − 3x + 2 = x − 1

Rearrange: x² − 4x + 3 = 0. The discriminant is Δ = (−4)² − 4 × 1 × 3 = 4 > 0, so there are two intersections. Factoring gives (x − 1)(x − 3) = 0, so x = 1 or x = 3. Substituting into y = x − 1 gives y = 0 and y = 2. The intersection points are (1, 0) and (3, 2).

整理得 x² − 4x + 3 = 0。判别式为 Δ = (−4)² − 4 × 1 × 3 = 4 > 0，因此有两个交点。因式分解得 (x − 1)(x − 3) = 0，所以 x = 1 或 x = 3。代入 y = x − 1 得 y = 0 和 y = 2。交点坐标为 (1, 0) 和 (3, 2)。

5. Quadratic-Quadratic Intersections | 二次函数之间的交点

For two quadratic functions, set the expressions equal and move all terms to one side. If the x² terms have the same coefficient, they cancel and the equation becomes linear.

对于两个二次函数，先令两个表达式相等，再把所有项移到同一边。如果 x² 项系数相同，那么 x² 会消去，方程变成一次方程。

a₁x² + b₁x + c₁ = a₂x² + b₂x + c₂

If the x² terms do not cancel, the result is a quadratic equation. Again, use the discriminant to determine the number of intersection points.

如果 x² 项没有消去，结果就是一个二次方程。此时仍可用判别式判断交点个数。

Example: Find the intersection of y = x² + 2x and y = x² − 4.

例：求 y = x² + 2x 与 y = x² − 4 的交点。

x² + 2x = x² − 4

The x² terms cancel, leaving 2x = −4, so x = −2. Substituting back gives y = (−2)² + 2(−2) = 0. The intersection point is (−2, 0).

x² 项消去后得到 2x = −4，因此 x = −2。代回原式得 y = (−2)² + 2(−2) = 0。交点坐标为 (−2, 0)。

6. Rational Functions and Domain Restrictions | 分式函数与定义域限制

Rational functions have the form f(x) = P(x)/Q(x), where Q(x) cannot be zero. When solving equations involving fractions, multiply through by the denominator, but always check the domain at the end.

分式函数的形式为 f(x) = P(x)/Q(x)，其中 Q(x) 不能为零。在解含分式的方程时，可以两边乘以分母，但最后必须检查定义域。

Example: Find the intersections of y = 1/(x − 2) and y = x − 3.

例：求 y = 1/(x − 2) 与 y = x − 3 的交点。

1/(x − 2) = x − 3

Multiply both sides by (x − 2):

两边同时乘以 (x − 2)：

1 = (x − 2)(x − 3)

Expanding gives x² − 5x + 6 = 1, so x² − 5x + 5 = 0. Using the quadratic formula, x = (5 ± √5)/2. Neither value equals 2, so both are valid. The corresponding y-values are found by substituting into y = x − 3.

展开得 x² − 5x + 6 = 1，即 x² − 5x + 5 = 0。由求根公式得 x = (5 ± √5)/2。两个值都不等于 2，因此都满足定义域。把 x 值代入 y = x − 3 即可求出对应的 y 坐标。

Always check for extraneous roots: a value that makes a denominator zero must be rejected.

一定要检查增根：任何使分母为零的 x 值都必须舍去。

7. Intersections with the Coordinate Axes | 与坐标轴的交点

Intersections with the y-axis are found by setting x = 0. Intersections with the x-axis are found by setting y = 0.

与 y 轴的交点通过令 x = 0 求出；与 x 轴的交点通过令 y = 0 求出。

For y = f(x), the y-intercept is f(0).

对于 y = f(x)，y 截距为 f(0)。
For x-intercepts, solve f(x) = 0.

求 x 截距时，解方程 f(x) = 0。

In IB questions, a point where the graph crosses the x-axis is often called a zero or root of the function. These are also the x-coordinates of the intersection between y = f(x) and the line y = 0.

在 IB 题目中，图像与 x 轴交点的横坐标通常称为函数的零点或根。这些点其实也是 y = f(x) 与直线 y = 0 的交点。

8. Graphical Estimation and Sketching | 画图估算与草图检验

A sketch can help you estimate the number of intersections and check whether your algebraic answer is reasonable. It is especially useful before solving a difficult equation.

画草图可以帮助你估计交点个数，并检验代数答案是否合理。在解较复杂的方程前先画图尤其有用。

Mark the x- and y-intercepts of each function.

标出每个函数的 x 截距和 y 截距。
For rational functions, draw vertical asymptotes and horizontal asymptotes.

对于分式函数，画出垂直渐近线和水平渐近线。
For quadratics, identify the vertex and direction of opening.

对于二次函数，确定顶点位置和开口方向。
Use the sketch to estimate where the curves cross.

利用草图大致判断曲线在哪些位置相交。

However, a sketch alone is never enough for exact coordinates. Always use algebra to confirm the x-values.

但是，仅凭草图永远无法得到精确坐标。必须用代数方法确认 x 值。

9. Using Technology | 使用图形计算器

In IB assessments where a calculator is allowed, modern GDCs have an intersection command. It finds the coordinates of the intersection points numerically.

在允许使用计算器的 IB 考试中，图形计算器通常带有求交点功能，可以数值化地直接求出交点坐标。

Follow these steps:

操作步骤如下：

Enter both functions as separate equations in Y1 and Y2.

在 Y1 和 Y2 中分别输入两个函数。
Choose a window that shows the relevant region.

选择合适的窗口范围，使相关区域可见。
Use the intersect or solve function to find each intersection.

使用 intersect 或 solve 功能找出每个交点。
Write down the exact or approximate coordinates as required by the question.

根据题目要求写出精确值或近似坐标。

Graphical calculators are efficient, but you should still understand the underlying algebra. In Paper 1, where calculators are not allowed, you must solve intersections by hand.

图形计算器效率很高，但你仍然需要理解背后的代数原理。在不能使用计算器的 Paper 1 中，你必须手算交点。

10. Special Cases: Tangency, No Solution and Identical Functions | 特殊情形：相切、无解与函数重合

Not every pair of graphs has a clear set of intersection points. Some situations require extra care.

并非每一对图像都有明确的交点。有些特殊情况需要额外小心。

Tangent: When the discriminant is zero, the two graphs touch at exactly one point. The line is tangent to the curve.

相切：当判别式等于零时，两个图像只接触一个点，此时直线与曲线相切。
No intersection: When the equation has no real solution, the graphs do not meet in the real plane.

无交点：当方程没有实数解时，两个图像在实数平面内不相交。
Identical functions: If f(x) = g(x) for every value of x, then the two graphs coincide and there are infinitely many common points.

函数重合：如果 f(x) = g(x) 对所有 x 都成立，那么两个图像完全重合，公共点有无数个。

For example, the equation x² = −1 has no real solution, so the graphs of y = x² and y = −1 do not intersect.

例如，方程 x² = −1 没有实数解，因此 y = x² 与 y = −1 的图像不会相交。

11. Applying Intersections to Real-Life Problems | 交点在实际问题中的应用

Intersection points are often used in modelling. The break-even point in business, for instance, is the point where cost and revenue are equal.

交点在数学建模中非常常用。例如，商业中的盈亏平衡点就是成本和收入相等时的点。

Example: A company’s cost is C(x) = 50 + 2x and its revenue is R(x) = 5x. Break-even occurs when C(x) = R(x).

例：某公司成本为 C(x) = 50 + 2x，收入为 R(x) = 5x。盈亏平衡时满足 C(x) = R(x)。

50 + 2x = 5x

This gives 3x = 50, so x = 50/3 ≈ 16.7. The two

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Methods for Finding Intersections of Function Graphs | 求函数图像交点的方法