Name: finding points on the coordinate plane
Uploaded: 2025-11-26T05:44:24.000Z
Duration: 1 min 21 s
Channel: JoVE

Płaszczyzna współrzędnych, czyli płaszczyzna kartezjańska, składa się z dwóch prostopadłych osi liczbowych: poziomej osi x i pionowej osi y.Osie te przecinają się w punkcie początkowym układu współrzędnych — punkcie, w którym obie wartości są równe zero — dzieląc płaszczyznę na cztery oznaczone ćwiartki.Uporządkowana para współrzędnych reprezentuje każde położenie na płaszczyźnie. Pierwsza liczba wskazuje położenie wzdłuż osi x, a druga pokazuje pozycję na osi y.Współrzędne działają jak system adresów, jednoznacznie identyfikując każdy punkt na płaszczyźnie.Na przykład punkt o współrzędnych (2, 3) leży dwie jednostki po prawej stronie i trzy jednostki powyżej początku układu współrzędnych.Wszystkie punkty z dodatnimi wartościami x leżą po prawej stronie osi y, a te z ujemnymi wartościami x leżą po lewej stronie. Linia pozioma reprezentuje punkty o tej samej wartości y, a linia pionowa pokazuje punkty o tej samej wartości x — dodatniej lub ujemnej.Układ współrzędnych ma zastosowanie w świecie rzeczywistym. Na przykład drony polegają na nim, aby podążać precyzyjnymi trasami lotu i docierać do określonych lokalizacji.

układ współrzędnych

Kartezja&#324;ski uk&#322;ad wsp&#243;&#322;rz&#281;dnych to fundamentalna struktura matematyczna, kt&#243;ra umo&#380;liwia wizualizacj&#281; zwi&#261;zk&#243;w mi&#281;dzy warto&#347;ciami liczbowymi w dw&#243;ch wymiarach. Sk&#322;ada si&#281; z dw&#243;ch przecinaj&#261;cych si&#281; osi liczbowych: poziomej osi x i pionowej osi y. Osie te przecinaj&#261; si&#281; w pocz&#261;tku uk&#322;adu wsp&#243;&#322;rz&#281;dnych, czyli w punkcie, w kt&#243;rym obie warto&#347;ci s&#261; r&#243;wne zeru. Ich przeci&#281;cie dzieli p&#322;aszczyzn&#281; na cztery kwadranty, oznaczone w kierunku przeciwnym do ruchu wskaz&#243;wek zegara, zaczynaj&#261;c od prawego g&#243;rnego.
Uporz&#261;dkowana para liczb reprezentuje ka&#380;dy punkt w uk&#322;adzie wsp&#243;&#322;rz&#281;dnych. Pierwsza liczba, wsp&#243;&#322;rz&#281;dna x, wskazuje po&#322;o&#380;enie na osi poziomej, a druga liczba, wsp&#243;&#322;rz&#281;dna y, wskazuje po&#322;o&#380;enie na osi pionowej. Wsp&#243;&#322;rz&#281;dne te funkcjonuj&#261; jak system adresowy, jednoznacznie okre&#347;laj&#261;c po&#322;o&#380;enie ka&#380;dego punktu. Na przyk&#322;ad punkt o wsp&#243;&#322;rz&#281;dnych (2, 3) le&#380;y dwie jednostki na prawo i trzy jednostki powy&#380;ej pocz&#261;tku uk&#322;adu wsp&#243;&#322;rz&#281;dnych, w pierwszym kwadrancie. Ujemne warto&#347;ci x powoduj&#261;, &#380;e punkty znajduj&#261; si&#281; po lewej stronie osi pionowej, a ujemne warto&#347;ci y &#8211; poni&#380;ej osi poziomej.
Punkty o tej samej warto&#347;ci y le&#380;&#261; wzd&#322;u&#380; prostej poziomej, a te o tej samej warto&#347;ci x, wzd&#322;u&#380; prostej pionowej. Linie te mog&#261; r&#243;wnie&#380; s&#322;u&#380;y&#263; jako granice, definiuj&#261;c okre&#347;lone obszary p&#322;aszczyzny. Na przyk&#322;ad wszystkie punkty mi&#281;dzy dwiema prostymi poziomymi tworz&#261; poziomy pas, podczas gdy pionowy pas powstaje mi&#281;dzy dwiema prostymi pionowymi.
Ten system pozwala na wyznaczanie pojedynczych punkt&#243;w, konstruowanie figur geometrycznych i analiz&#281; danych. Stanowi podstaw&#281; dla r&#243;&#380;nych zastosowa&#324; w matematyce, nauce i technologii, gdzie reprezentacja przestrzenna i precyzja s&#261; kluczowe.

Układ współrzędnych

Kartezja&#324;ski uk&#322;ad wsp&#243;&#322;rz&#281;dnych to fundamentalna struktura matematyczna, kt&#243;ra umo&#380;liwia wizualizacj&#281; ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Coordinates and Graphs

Współrzędne i wykresy

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 x-coordinate, and it shows position along the horizontal axis. The second number is the y-coordinate, and it shows position along the vertical axis.

Ordered pairs work like an address system for the plane. For example, the point (2, 3) is two units to the right and three units above the origin. That point lies in the first quadrant. Negative x-values place points to the left of the vertical axis, and negative y-values place points below the horizontal axis.

Points with the same y-value lie on a horizontal line. Points with the same x-value lie on a vertical line. These lines can also mark boundaries for regions of the plane. A horizontal strip forms between two horizontal lines, and a vertical strip forms between two vertical lines.

The coordinate plane is useful for plotting individual points, building geometric shapes, and analyzing data. It gives a precise way to show relationships between numbers. This makes it important in mathematics, science, and technology, where clear spatial representation matters.

Finding Points on the Coordinate Plane

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

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 the Cartesian coordinate plane uses the x-axis, y-axis, quadrants, and ordered pairs to plot points, draw shapes, and show data clearly.

Finding Points on the Coordinate Plane | 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

Finding Points on the Coordinate Plane

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