Using Graphs to Solve other Equations | 使用图像解其他方程

📚 Using Graphs to Solve other Equations | 使用图像解其他方程

Graphs are not just visual representations of relationships; they are powerful problem-solving tools. When an equation is difficult to solve algebraically, or when we need to understand its solutions in context, drawing or reading a graph can give us the answer directly. This revision guide explores how to use graphs to solve equations of various types, from linear and quadratic to cubic and simultaneous equations.

图像不仅仅是关系的视觉表示，更是强大的解题工具。当方程难以用代数方法求解，或者我们需要在情境中理解其解时，绘制或读取图像可以直接给出答案。本复习指南探讨如何利用图像解各类方程，包括线性、二次、三次以及联立方程。

1. Reading Solutions from a Graph | 从图像中读取解

When a graph of a function y = f(x) is drawn, the solutions to the equation f(x) = 0 are exactly the x-coordinates where the curve crosses the x-axis. These points are called the roots or x-intercepts. To solve f(x) = k, we look for the x-coordinates where the horizontal line y = k meets the curve.

当函数 y = f(x) 的图像被画出时，方程 f(x) = 0 的解正是曲线与 x 轴交点的 x 坐标。这些点称为根或 x 轴截距。若要解 f(x) = k，我们需找到水平线 y = k 与曲线交点的 x 坐标。

To solve f(x) = 0, read the x-intercepts of y = f(x).

要解 f(x) = 0，读取 y = f(x) 的 x 轴截距。
To solve f(x) = k, draw the horizontal line y = k and read the x-coordinates of all intersection points.

要解 f(x) = k，画出水平线 y = k，并读取所有交点的 x 坐标。
If the curve does not cross the x-axis, the equation f(x) = 0 has no real solutions.

如果曲线不穿过 x 轴，则方程 f(x) = 0 没有实数解。

2. Solving Linear Equations Using Graphs | 用图像解线性方程

A linear equation such as 2x + 1 = 5 can be solved graphically by drawing y = 2x + 1 and y = 5 on the same axes. The x-coordinate of their intersection is the solution. Alternatively, rearranging to y = 2x − 4 and finding its x-intercept gives the same answer.

线性方程如 2x + 1 = 5 可通过在同一坐标轴上绘制 y = 2x + 1 和 y = 5 来求解。它们交点的 x 坐标即为解。或者，将其重排为 y = 2x − 4 并求其 x 轴截距，也会得到相同的答案。

2x + 1 = 5 ⇒ x = 2

For a straight line, one unique solution exists. The graphical method is especially useful when the equation is not neatly solvable by inspection, or when checking a result.

对于直线，存在唯一解。图像法在方程不易直接观察求解或需要验证结果时尤其有用。

3. Solving Quadratic Equations Using Graphs | 用图像解二次方程

A quadratic equation ax² + bx + c = 0 corresponds to the x-intercepts of the parabola y = ax² + bx + c. A parabola may cross the x-axis twice, touch it once, or not at all, giving two, one, or zero real roots respectively.

二次方程 ax² + bx + c = 0 对应于抛物线 y = ax² + bx + c 的 x 轴截距。抛物线可能与 x 轴交于两点、相切于一点或完全不相交，分别对应两个、一个或零个实数根。

Two distinct roots: the curve crosses the x-axis at two different points.
One repeated root: the vertex touches the x-axis.
No real roots: the curve stays entirely above or below the x-axis.

两个不同实根：曲线在两点穿过 x 轴。
一个重根：顶点与 x 轴相切。
无实根：曲线完全在 x 轴上方或下方。

To solve x² − 4x + 3 = 0, plot y = x² − 4x + 3. The x-intercepts are x = 1 and x = 3, confirming the factorised form (x − 1)(x − 3) = 0.

要解 x² − 4x + 3 = 0，绘制 y = x² − 4x + 3。x 截距为 x = 1 和 x = 3，这与因式分解形式 (x − 1)(x − 3) = 0 一致。

4. Solving Cubic and Higher-Degree Equations | 解三次及更高次方程

Cubic equations such as x³ − 3x + 1 = 0 are often difficult to solve algebraically. By plotting y = x³ − 3x + 1, we read the x-coordinates where the curve crosses the x-axis. A cubic graph can have one, two, or three real roots.

三次方程如 x³ − 3x + 1 = 0 通常难以代数求解。通过绘制 y = x³ − 3x + 1，我们读取曲线与 x 轴交点的 x 坐标。三次图像可以有一个、两个或三个实数根。

At each crossing, the y-value is zero. Therefore, if the curve crosses at x ≈ −1.879, x ≈ 0.347, and x ≈ 1.532, these are the approximate solutions of the equation. For higher-degree polynomials, the same principle applies: roots are the x-intercepts of the graph.

在每个交点处，y 值为零。因此，如果曲线在 x ≈ −1.879、x ≈ 0.347 和 x ≈ 1.532 处穿过 x 轴，这些就是方程的近似解。对于更高次多项式，同样原理适用：根就是图像的 x 截距。

5. Rearranging Equations to Use a Given Graph | 重排方程以利用给定图像

Often a graph of y = f(x) is given, and we must solve a different equation g(x) = 0. If g(x) can be written as f(x) − h(x) = 0, then solutions are the x-coordinates where f(x) intersects y = h(x). We can draw the line or curve y = h(x) on the same axes.

通常给定 y = f(x) 的图像，而我们需要解另一个方程 g(x) = 0。如果 g(x) 可以写成 f(x) − h(x) = 0 的形式，那么解就是 f(x) 与 y = h(x) 交点的 x 坐标。我们可以在同一坐标系中画出直线或曲线 y = h(x)。

For example, given the graph y = x², to solve x² − 2x − 3 = 0, rearrange to x² = 2x + 3. The solutions are the x-coordinates of the intersections of y = x² and y = 2x + 3, giving x = −1 and x = 3.

例如，给定 y = x² 的图像，要解 x² − 2x − 3 = 0，可重排为 x² = 2x + 3。解是 y = x² 与 y = 2x + 3 交点的 x 坐标，得到 x = −1 和 x = 3。

6. Solving Simultaneous Equations Graphically | 用图像解联立方程

The solution to two simultaneous equations corresponds to the point(s) where their graphs intersect. For a linear equation and a quadratic equation, the line may cut the parabola at two points, touch it at one point, or miss it entirely.

两个联立方程的解对应其图像的交点。对于线性方程和二次方程，直线可能与抛物线相交于两点、相切于一点或完全不相交。

y = 2x + 1 and y = x² − 2x + 3

Substituting gives x² − 4x + 2 = 0, which has two solutions. Graphically, the line and parabola meet at two points. Reading the coordinates gives approximate solutions, e.g. (0.586, 2.172) and (3.414, 7.828).

代入得到 x² − 4x + 2 = 0，有两个解。图像上，直线和抛物线相交于两点。读取坐标得到近似解，例如 (0.586, 2.172) 和 (3.414, 7.828)。

7. Solving Equations of the Form f(x) = g(x) | 解 f(x) = g(x) 形式的方程

Any equation of the form f(x) = g(x) can be solved by plotting both functions on the same axes. The x-coordinates of every intersection point are the solutions. This avoids the need to rearrange the equation into f(x) − g(x) = 0, although that is also a valid approach.

任何 f(x) = g(x) 形式的方程都可以通过在同一坐标轴上绘制两个函数的图像来求解。每个交点的 x 坐标就是解。这避免了将方程重排为 f(x) − g(x) = 0 的需要，尽管后者也是有效的方法。

This method is widely used in numerical approximation, especially when equations involve transcendental functions such as sin x or 2ˣ. The graphical intersection can then be refined using other techniques.

该方法广泛用于数值近似，尤其是当方程涉及超越函数如 sin x 或 2ˣ 时。之后可以再用其他技术细化图像交点。

8. Using Graphs to Solve Inequalities | 用图像解不等式

Graphs can also solve inequalities. To solve f(x) > 0, identify the intervals where the graph of y = f(x) lies above the x-axis. To solve f(x) < g(x), find the x-values where the graph of f lies below the graph of g.

图像也可以解不等式。要解 f(x) > 0，找出 y = f(x) 的图像位于 x 轴上方的区间。要解 f(x) < g(x)，找出 f 的图像位于 g 的图像下方的 x 值区间。

Inequality	Graphical View
f(x) > 0	Curve above the x-axis
f(x) < 0	Curve below the x-axis
f(x) ≥ g(x)	Curve of f at or above curve of g

Remember to pay attention to strict and non-strict inequalities when writing the final answer.

写最终答案时，务必注意严格不等式和非严格不等式的区别。

9. Drawing Accurate Graphs by Plotting Points | 通过描点绘制精确图像

An accurate graph is essential for reliable solutions. To plot a curve, choose a table of x-values, calculate the corresponding y-values, plot the points, and join them with a smooth curve. A straight-line graph needs only two points, but a curve needs several well-chosen points.

精确的图像对于获得可靠解至关重要。要绘制曲线，先选 x 值表格，计算对应 y 值，描点，然后用平滑曲线连接。直线图只需两个点，但曲线需要多个精心选择的点。

For quadratics, include points around the turning point. For cubics, include points on both sides of any local maximum or minimum. When solving equations, label axes, intersection points, and roots clearly.

对于二次函数，应包括顶点附近的点。对于三次函数，应包括局部极大值或极小值两侧的点。在解方程时，要清晰标注坐标轴、交点和根。

10. Solving Exponential and Trigonometric Equations | 用图像解指数和三角方程

Equations involving eˣ, 2ˣ, sin x, or cos x often have no simple algebraic solution. By graphing y = 2ˣ and y = 3x + 1, the intersection gives an approximate solution to 2ˣ = 3x + 1. Similarly, to solve sin x = x − 1, plot both curves.

涉及 eˣ、2ˣ、sin x 或 cos x 的方程通常没有简单的代数解。通过绘制 y = 2ˣ 和 y = 3x + 1，它们的交点给出 2ˣ = 3x + 1 的近似解。类似地，要解 sin x = x − 1，可绘制两条曲线。

Trigonometric equations may have infinitely many solutions. When using a graph, restrict the domain to the range specified in the question, and read all intersections within that interval.

三角方程可能有无限多个解。使用图像时，应将定义域限制在题目指定的范围内，并读取该区间内的所有交点。

11. Solving Equations Involving Reciprocals | 用图像解含倒数形式的方程

Reciprocal graphs such as y = 1/x have vertical asymptotes where the function is undefined. To solve equations like 1/x = x − 2, plot y = 1/x and y = x − 2. The intersections give solutions; here, the line meets the hyperbola at two points, corresponding to the roots of x² − 2x − 1 = 0.

倒数图像如 y = 1/x 在函数无定义处有垂直渐近线。要解 1/x = x − 2 这类方程，可绘制 y = 1/x 和 y = x − 2。交点给出解；此处直线与双曲线相交于两点，对应 x² − 2x − 1 = 0 的根。

Be careful not to include the x-value where the original fraction is undefined as a solution.

注意不要将原分数无定义处的 x 值包含在解中。

12. Limitations and Accuracy | 局限性与精确度

Graphical methods often yield only approximate solutions, limited by the precision of drawing and reading. A small error in plotting can lead to a significant error in the root. For exact answers, algebraic or numerical methods such as iterative solving should be used.

图像法通常只能得到近似解，其精度受绘图和读图限制。描点时的微小误差可能导致根的显著误差。要获得精确答案，应使用代数法或迭代等数值方法。

However, graphs are invaluable for checking answers, finding initial estimates, and understanding the number and nature of solutions. In examinations, show clearly how you read the graph and give answers to a sensible degree of accuracy.

然而，图像在验证答案、寻找初始估计、理解解的数量和性质方面无可替代。在考试中，请清晰展示你如何从图像中读数，并给出合理的精度。

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Using Graphs to Solve other Equations | 使用图像解其他方程