📚 Finding Where Graphs Meet | 求函数图像的交点
In IB Mathematics, one of the fundamental skills is finding the coordinates where two graphs intersect. This process, often called solving simultaneous equations, underpins many topics from linear functions to quadratics, circles, and beyond. Whether you are using algebra or a graphing calculator, understanding how to find where graphs meet helps you analyse systems and model real-world situations.
在 IB 数学中，一项基本技能是求两个函数图像相交的坐标。这个过程通常被称为解联立方程，是线性函数、二次函数、圆乃至更多内容的基础。无论你使用代数方法还是图形计算器，理解如何寻找图形的交点有助于分析系统和建立现实世界的模型。
1. Introduction to Graph Intersections | 图形相交简介
where the line y = 2x + 1 meets the parabola y = x² - 3 involves solving 2x + 1 = x² - 3.
当两个函数绘制在同一坐标系上时，它们的交点就是同时满足两个方程的 (x, y) 值。例如，求直线 y = 2x + 1 与抛物线 y = x² - 3 的交点，就需要解方程 2x + 1 = x² - 3。
The number of intersections can be zero, one, two, or infinitely many, depending on the types of functions and their parameters. Graphically, you can see whether they cross, touch, or never meet.
交点的个数可能为零个、一个、两个或无穷多个，这取决于函数类型及其参数。从图形上看，你可以观察它们是相交、相切还是永不相交。
2. Solving Simultaneous Linear Equations | 解线性方程组
When both graphs are straight lines, their intersection is found by solving a system of two linear equations. The solution represents the unique point where the lines meet—unless the lines are parallel or coincident.
当两个图形均为直线时，通过解一个两元一次方程组即可找到它们的交点。这个解代表两直线相交的唯一一个点——除非这两条直线平行或重合。
There are two common algebraic methods: substitution and elimination. Both yield the same result, and choosing between them often depends on the form of the equations.
常用的代数方法有两种：代入法和消元法。两种方法都能得到相同的结果，选择哪种通常取决于方程的形式。
substitution and elimination quickly give x = 3, y = 1. The intersection point is (3, 1).
对于方程组 2x + y = 7 和 x - y = 2，无论是代入法还是消元法都可以迅速求得 x = 3, y = 1，交点为 (3, 1)。
If the lines are parallel, the slopes are equal but intercepts differ, so the system has no solution. If they are coincident, every point on the line is a solution—infinitely many intersections.
如果两直线平行，则斜率相等但截距不同，因此方程组无解。如果两直线重合，则直线上的每个点都是解——无穷多个交点。
It is crucial to check your solution by plugging the values back into both original equations. A common mistake is finding a solution that works for only one equation.
关键的一点是将所得数值代回原方程组进行检验。一个常见错误是找到的解只满足其中一个方程。
3. Substitution Method | 代入法
In the substitution method, you rearrange one equation to express one variable in terms of the other, and then substitute this expression into the second equation. This reduces the system to a single equation in one unknown.
在代入法中，你将其中一个方程变形，用含另一个变量的式子表示出其中一个变量，然后将该表达式代入另一个方程。这样可以将方程组转化为关于一个未知数的一元方程。
Example: y = 3x - 5 and 2x + y = 9
Substitute y → 2x + (3x - 5) = 9 → 5x - 5 = 9 → x = 2.8
Then y = 3(2.8) - 5 = 3.4 → Intersection (2.8, 3.4)
Substitution is particularly useful when one equation already has an isolated variable or can be easily rearranged. However, fractions can occasionally make the algebra messy.
当其中一个方程已经将某个变量单独表示出来或容易变形时，代入法尤其有用。不过，有时会出现分数，使得代数运算变得繁琐。
Always keep your work organised; clearly indicate the step where you perform the substitution to avoid errors.
始终保持解题步骤清晰；明确标出代入的步骤，以避免错误。
4. Elimination Method | 消元法
Elimination involves adding or subtracting the equations to cancel one variable. Often, you multiply one or both equations by constants to align coefficients.
消元法是通过将方程相加或相减来消去一个变量。通常需要将一个或两个方程乘以某个常数，使系数一致。
Consider 3x + 2y = 12 and 5x - 2y = 4
Adding eliminates y: 8x = 16 → x = 2, then y = 3 → (2,3)
Elimination is often cleaner for systems in standard form and avoids fractions. When coefficients are not easily aligned, you can multiply both equations to find a common multiple.
对于标准形式的方程组，消元法通常更简洁并能避免分数。当系数不易对齐时，可以两个方程都乘以适当的数以找到公倍数。
Always label your equations and keep track of which variable you are eliminating. Careless sign errors can lead to incorrect intersections.
记得给你的方程编号，并明确自己正在消去哪个变量。粗心的符号错误可能导致求出的交点出错。
5. Linear and Quadratic Intersections | 直线与二次函数的交点
When a straight line meets a quadratic curve, you equate the expressions for y and solve the resulting quadratic equation. The solutions for x can be found by factoring, completing the square, or using the quadratic formula.
当一条直线与二次曲线相交时，令两者的 y 表达式相等，然后解得到的二次方程。x 的解可以通过因式分解、配方法或使用求根公式求得。
Find intersections of y = x + 1 and y = x² - 2x - 3
Set x + 1 = x² - 2x - 3 → 0 = x² - 3x - 4 → (x - 4)(x + 1) = 0
x = 4 or x = -1 → Points (4,5) and (-1,0)
The number of intersections can be two, one (tangent), or zero—determined by the discriminant. Understanding this link between algebra and geometry is key for IB exam questions.
交点个数可能是两个、一个（相切）或零个——这由判别式决定。理解这种代数与几何之间的联系对于应对 IB 考试题目至关重要。
1 gives x² - x + 6 = 0; discriminant Δ = 1 - 24 = -23 < 0, so no real intersection exists.
例如，y = x - 5 和 y = x² + 1 得到 x² - x + 6 = 0；判别式 Δ = 1 - 24 = -23 < 0，因此不存在实数交点。
6. Using the Discriminant to Determine Number of Intersections | 利用判别式判断交点个数
x + c = 0, the discriminant Δ = b² - 4ac tells you how many times the line and curve meet.
当你令方程相等并形成形如 ax² + bx + c = 0 的二次方程后，判别式 Δ = b² - 4ac 可以告诉你直线与曲线相交的次数。
If Δ > 0 → two distinct intersection points
If Δ = 0 → line is tangent (one intersection)
If Δ < 0 → no real intersections
Example: Find k such that y = 2x + k is tangent to y = x² + 1.
示例：求 k 使得 y = 2x + k 与 y = x² + 1 相切。
Set x² + 1 = 2x + k → x² - 2x + (1 - k) = 0
Δ = (-2)² - 4(1)(1 - k) = 4 - 4 + 4k = 4k
For tangency, 4k = 0 → k = 0
This technique is frequently tested, especially in functions and equations questions. Always write the quadratic in standard form before identifying a, b, c.
这种技巧在考试中经常出现，尤其是在函数与方程类题目中。在识别 a、b、c 之前，务必先将二次方程化为标准形式。
7. Intersection of Circle and Line | 圆与直线的相交
Finding where a line meets a circle also leads to a quadratic equation, typically after substituting the linear equation into the circle's equation. The discriminant again reveals the number of intersections.
求一条直线与圆的交点也会导出一个二次方程，通常需要将直线方程代入圆的方程。同样，判别式揭示了交点的个数。
Circle: (x - 1)² + (y + 2)² = 25, Line: y = x + 3
Substitute: (x - 1)² + (x + 5)² = 25 → Expand and
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