Name: plotting two-variable equations with tables
Uploaded: 2025-11-26T05:44:21.000Z
Duration: 1 min 30 s
Channel: JoVE

包含两个变量的方程表示变量 x 和 y 之间的关系。该方程的解是满足该方程的有序数对 (x, y)。可以通过在图表上绘制相应的点来直观地表示这些溶液。绘图的过程始于选择若干个 x 的取值，并将其代入方程以计算出相应的 y 值。截距用于确定图像与坐标轴的交点。通过令 y 等于零可求得 x 轴截距，通过令 x 等于零可求得 y 轴截距。这些点将与其他已绘制的点一同添加到图中。将每个得到的有序数对记录下来，并在图中标出。然后将这些点连接起来，以显示出一个可视化的图形，该图形可能表现为一条直线或一条曲线，具体取决于方程的形式。这种绘图方法在现实情境中非常有用，例如追踪多天内的电影票销售情况。该图通过将每日售票量绘制成点，展示了销售额随时间的变化，其中 x 表示天数，y 表示总销售额。这种可视化模式有助于比较数值、识别趋势，并根据历史数据预测未来销售额。

二元方程的图像

Graphs of Equations in Two Variables

&#21547;&#26377;&#20004;&#20010;&#21464;&#37327;&#30340;&#26041;&#31243;&#36890;&#24120;&#20889;&#20316; y = f(x) &#25110; Ax + By = C&#65292;&#29992;&#20197;&#25551;&#36848;&#30001;&#21464;&#37327; x &#19982; y &#25152;&#20195;&#34920;&#30340;&#37327;&#20043;&#38388;&#30340;&#25968;&#37327;&#20851;&#31995;&#12290;&#35813;&#31867;&#26041;&#31243;&#30340;&#27599;&#19968;&#20010;&#35299;&#22343;&#20026;&#19968;&#20010;&#26377;&#24207;&#25968;&#23545; (x, y)&#65292;&#24403;&#23558;&#20854;&#20195;&#20837;&#26041;&#31243;&#21518;&#20351;&#31561;&#24335;&#25104;&#31435;&#12290;&#36890;&#36807;&#22270;&#24418;&#26041;&#24335;&#34920;&#31034;&#36825;&#20123;&#25968;&#23545;&#65292;&#21487;&#20197;&#30452;&#35266;&#22320;&#29702;&#35299;&#21464;&#37327;&#20043;&#38388;&#30340;&#20851;&#31995;&#12290;
&#32472;&#21046;&#20108;&#20803;&#26041;&#31243;&#22270;&#20687;&#30340;&#24120;&#29992;&#26041;&#27861;&#20043;&#19968;&#26159;&#24314;&#31435;&#25968;&#20540;&#23545;&#24212;&#34920;&#12290;&#39318;&#20808;&#36873;&#21462;&#33258;&#21464;&#37327; x &#30340;&#33509;&#24178;&#21462;&#20540;&#65292;&#24182;&#23558;&#36825;&#20123;&#20540;&#20195;&#20837;&#26041;&#31243;&#20197;&#35745;&#31639;&#30456;&#24212;&#30340;&#22240;&#21464;&#37327; y &#30340;&#21462;&#20540;&#12290;&#25152;&#24471;&#21040;&#30340; (x, y) &#25968;&#23545;&#21363;&#20026;&#31515;&#21345;&#23572;&#22352;&#26631;&#24179;&#38754;&#19978;&#30340;&#28857;&#12290;&#23558;&#36825;&#20123;&#28857;&#32472;&#21046;&#24182;&#36830;&#32447;&#21518;&#65292;&#20415;&#21487;&#25581;&#31034;&#26041;&#31243;&#25152;&#34920;&#31034;&#20851;&#31995;&#30340;&#24615;&#36136;&#8212;&#8212;&#26080;&#35770;&#26159;&#32447;&#24615;&#12289;&#20108;&#27425;&#22411;&#36824;&#26159;&#20854;&#20182;&#31867;&#22411;&#12290;
&#20363;&#22914;&#65292;&#23545;&#20110;&#32447;&#24615;&#26041;&#31243; y = 2x + 1&#65292;&#24403;&#36873;&#25321;&#22810;&#20010; x &#20540;&#35745;&#31639;&#30456;&#24212;&#30340; y &#20540;&#26102;&#65292;&#21487;&#24471;&#65306;
<ul>
<li>&#33509; x = -1&#65292;&#21017; y = 2(-1) + 1 = -1</li>
<li>&#33509; x = 0&#65292;&#21017; y = 1</li>
<li>&#33509; x = 2&#65292;&#21017; y = 5</li>
</ul>
&#23558;&#36825;&#20123;&#28857; (-1, -1)&#12289;(0, 1) &#19982; (2, 5) &#32472;&#21046;&#22312;&#22352;&#26631;&#24179;&#38754;&#19978;&#24182;&#21152;&#20197;&#36830;&#25509;&#65292;&#21363;&#24418;&#25104;&#19968;&#26465;&#30452;&#32447;&#65292;&#20195;&#34920;&#35813;&#26041;&#31243;&#30340;&#20840;&#37096;&#35299;&#12290;
&#25130;&#36317;&#26159;&#20998;&#26512;&#22270;&#20687;&#29305;&#24449;&#30340;&#37325;&#35201;&#21442;&#32771;&#28857;&#12290;&#20196; y = 0&#65292;&#35299;&#24471; x &#25130;&#36317;&#65307;&#20196; x = 0&#65292;&#35299;&#24471; y &#25130;&#36317;&#12290;&#36825;&#20004;&#20010;&#28857;&#20998;&#21035;&#34920;&#31034;&#22270;&#20687;&#19982; x &#36724;&#21450; y &#36724;&#30340;&#20132;&#28857;&#12290;&#20197;&#26041;&#31243; y = 2x + 1 &#20026;&#20363;&#65292;&#35774; x = 0 &#26102;&#24471; y &#25130;&#36317; (0, 1)&#65292;&#35774; y = 0 &#26102;&#24471; x &#25130;&#36317; (-0.5, 0)&#12290;&#36825;&#20123;&#25130;&#36317;&#26377;&#21161;&#20110;&#26356;&#28145;&#20837;&#22320;&#29702;&#35299;&#26041;&#31243;&#22270;&#20687;&#30340;&#32467;&#26500;&#19982;&#21464;&#21270;&#35268;&#24459;&#12290;

&#21547;&#26377;&#20004;&#20010;&#21464;&#37327;&#30340;&#26041;&#31243;&#36890;&#24120;&#20889;&#20316; y = f(x) &#25110; Ax + By = ...

Education

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JoVE Core

Math Fundamentals

数学基础

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Coordinates and Graphs

KEY TERMS AND CONCEPTS

关键术语与概念

plotting two-variable equations with tables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation.

A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian coordinate plane.

After the points are plotted, connect them to see the shape of the relationship. Some equations make a straight line, while others may form a quadratic or another curve. For example, the linear equation y = 2x + 1 gives points such as (-1, -1), (0, 1), and (2, 5).

Intercepts help identify where the graph crosses the axes. The x-intercept is found by setting y = 0 and solving for x. The y-intercept is found by setting x = 0 and solving for y. For y = 2x + 1, the y-intercept is (0, 1) and the x-intercept is (-0.5, 0).

Plotting Two-Variable Equations with Tables

plotting-two-variable-equations-with-tables

coordinate plane

Finding Points on the Coordinate Plane

The Cartesian coordinate plane is used to find and plot points in two dimensions. It is made of two number lines that cross at right angles. The horizontal line is the x-axis, and the vertical line is the y-axis. Their meeting point is the origin, where both values are zero. The axes divide the plane into four quadrants. These quadrants are labeled in a counterclockwise direction, starting from the upper right. Every point on the plane is written as an ordered pair. The first number is the...

the distance formula

Finding Distance on a Coordinate Plane

Finding distance on a coordinate plane uses the distance formula and the Pythagorean Theorem. In geometry, this method helps measure the direct distance between two points on a flat plane. It is useful in many practical and theoretical settings. The formula comes from a right triangle. On a coordinate plane, the horizontal and vertical differences between two points become the legs of that triangle. The line connecting the points is the hypotenuse, and that is the shortest path between them. <br...

the midpoint formula

Finding the Center of a Line Segment

Finding the center of a line segment is a key idea in coordinate geometry. The midpoint is the point that lies exactly halfway between two points on a coordinate plane. It is used in mathematics, physics, engineering, and planning work where a central location matters. To work with a midpoint, start with two points, A(x1, y1) and B(x2, y2). A straight line segment can be drawn between them on the coordinate plane. The midpoint, labeled M, divides that segment into two equal parts. The...

graphs of equations in two variables

Plotting two-variable equations with tables helps show how x and y are related. An equation written as y = f(x) or Ax + By = C gives ordered pairs that satisfy the rule. Each pair can be checked by substituting the x and y values into the equation. A value table is a common way to build the graph. Start by choosing several values for the independent variable x. Then substitute each x value into the equation to find the matching y value. The results become ordered pairs on the Cartesian...

circles

Circle Equations in Standard Form

Circle equations in the coordinate plane use a center and a radius to describe every point on the curve. A circle is the set of all points that stay the same distance from a fixed point called the center. That constant distance is the radius. The distance formula connects a point x, y on the circle to the center h, k. The distance between those two points equals the radius r. When both sides of the distance formula are squared, the equation of the circle can be written in standard form. <br...

symmetry

Ellipse Symmetry at the Origin

An ellipse centered at the origin shows clear symmetry in its graph. Its equation describes a smooth, closed curve whose distance from the center keeps a constant ratio between the horizontal and vertical axes. Because the ellipse stretches farther along the x-axis than the y-axis, it has a horizontal shape. This ellipse has symmetry across the x-axis. That means the top half and bottom half are mirror images across the horizontal axis. Algebraically, the equation stays the same when y is...

solving equations graphically

Graphing Equations to Find Intercepts

Graphing equations helps students find intercepts and solutions on the coordinate plane. It gives a visual way to solve equations and understand how a function behaves. To use this method, write the equation in the form y = f(x). The solution to the original equation is the x-value where the graph crosses the x-axis. At that point, f(x) = 0. For example, the linear equation 2x - 4 = 0 can be written as y = 2x - 4. When this function is graphed, it has one x-intercept at x = 2, and that is the...

solving inequalities graphically

Graphing Inequality Solution Ranges

Graphing inequality solution ranges gives a visual way to find the x-values that make an inequality true. Students look at where a graph sits compared with the x-axis or with another graph. This shows the interval where the condition is satisfied. For a quadratic inequality, the first step is to plot the related function. For example, with x² − 4x + 2 &lt; 0, graph y = x² − 4x + 2. The solution is the set of x-values where the curve lies below the x-axis. Inequalities can also compare two...

key terms and concepts

Learn how to graph equations in two variables using value tables, ordered pairs, and intercepts for clear high school math practice.

Plotting Two-Variable Equations with Tables | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Plotting Two-Variable Equations with Tables

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