What is a y-coordinate?

The y-coordinate is an essential concept in mathematics, particularly in coordinate geometry. It represents the vertical position of a point in a two-dimensional plane. To understand it fully, let’s break down the idea step by step.

Coordinate System

Cartesian Coordinate System

The most common coordinate system is the Cartesian coordinate system, named after the French mathematician René Descartes. This system uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is called the origin, denoted by (0,0).

Points in the Plane

Any point in the Cartesian plane can be represented by an ordered pair of numbers (x, y). The first number, x, is the x-coordinate, and it tells us how far to move horizontally from the origin. The second number, y, is the y-coordinate, and it tells us how far to move vertically from the origin.

For example, consider the point (3, 4):

  • The x-coordinate is 3, so we move 3 units to the right from the origin.
  • The y-coordinate is 4, so we move 4 units up from there.

Importance of the y-coordinate

Determining Position

The y-coordinate is crucial for determining the exact position of a point in the plane. For instance, if you know the coordinates of a point are (5, 7), you can immediately tell that the point is 5 units to the right and 7 units up from the origin.

Graphing Functions

In graphing functions, the y-coordinate often represents the output of a function for a given x-coordinate (input). For example, in the function $y = 2x + 3$, if $x = 1$, then $y = 2(1) + 3 = 5$. Here, the point (1, 5) is plotted on the graph.

Real-World Applications

The concept of the y-coordinate is widely used in various fields, including physics, engineering, and computer science. For example, in physics, the y-coordinate can represent the height of an object above the ground, while the x-coordinate represents its horizontal position.

Examples and Practice

Example 1: Plotting Points

Let’s plot the following points on a Cartesian plane: (2, 3), (-1, 4), (0, -2), and (-3, -3).

  1. For the point (2, 3): Move 2 units to the right and 3 units up.
  2. For the point (-1, 4): Move 1 unit to the left and 4 units up.
  3. For the point (0, -2): Stay at the origin horizontally and move 2 units down.
  4. For the point (-3, -3): Move 3 units to the left and 3 units down.

Example 2: Understanding Functions

Consider the function $y = x^2 – 4x + 6$. To graph this function, we can calculate the y-coordinates for various x-values.

  1. If $x = 0$, then $y = 0^2 – 4(0) + 6 = 6$. Plot the point (0, 6).
  2. If $x = 2$, then $y = 2^2 – 4(2) + 6 = 2$. Plot the point (2, 2).
  3. If $x = -1$, then $y = (-1)^2 – 4(-1) + 6 = 11$. Plot the point (-1, 11).

By plotting several points and connecting them, you can visualize the function’s graph.

Conclusion

The y-coordinate is a fundamental aspect of coordinate geometry, enabling us to describe the vertical position of points in a plane. It plays a vital role in graphing functions and has numerous real-world applications. By understanding how to work with y-coordinates, you can solve various mathematical problems and better appreciate the beauty of geometry.

1. Wikipedia – Cartesian Coordinate System

Citations

  1. 2. Khan Academy – Coordinate Plane
  2. 3. Math is Fun – Cartesian Coordinates

Related

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H + HO2 → O2 + H2 k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O2 k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) H + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-5 s^-1) φ

Table 1 Reactions, rate constants and activation energies used in the model* No. Reaction kopt (M⁻¹ s⁻¹) 1 OH + H₂ → H + H₂O 3.74 x 10⁷ 2 OH + HO₂ → HO₂ + OH⁻ 5 x 10⁹ 3 OH + H₂O₂ → HO₂ + H₂O 3.8 x 10⁷ 4 OH + O₂ → O₂ + OH 9.96 x 10⁹ 5 OH + HO₂ → O₂ + H₂O 7.1 x 10⁹ 6 OH + OH → H₂O₂ 5.3 x 10⁹ 7 OH + e⁻aq → OH⁻ 3 x 10¹⁰ 8 H + O₂ → HO₂ 2.0 x 10¹⁰ 9 H + HO₂ → H₂O₂ 2.0 x 10¹⁰ 10 H + H₂O₂ → OH + H₂O 3.44 x 10⁷ 11 H + OH → H₂O 1.4 x 10¹⁰ 12 H + H → H₂ 1.94 x 10¹⁰ 13 e⁻aq + O₂ → O₂⁻ 1.9 x 10¹⁰ 14 e⁻aq + O₂ → HO₂⁻ + OH⁻ 1.3 x 10¹⁰ 15 e⁻aq + HO₂ 2.0 x 10¹⁰ 16 e⁻aq + H₂O₂ 1.1 x 10¹⁰ 17 e⁻aq + HO₂ → OH + OH⁻ 1.3 x 10¹⁰ 18 e⁻aq + H⁺ → H 2.3 x 10¹⁰ 19 e⁻aq + e⁻aq → H₂ + OH⁻ + OH⁻ 2.5 x 10⁹ 20 HO₂ + O₂ → O₂ + HO₂ 1.3 x 10⁹ 21 HO₂ + HO₂ → O₂ + H₂O₂ 8.3 x 10⁵ 22 HO₂ + HO₂ → O₂ + OH + H₂O 3.7 23 HO₂ + HO₂ → O₂ + O₂ + OH + H₂O 7 x 10⁵ s⁻¹ 24 H⁺ + O₂⁻ → HO₂ 4.5 x 10¹⁰ 25 H⁺ + O₂⁻ → O₂ 2.0 x 10¹⁰ 26 H⁺ + OH⁻ 1.4 x 10¹¹ 27 H⁺ + HO₂⁻ 2 x 10¹⁰ 28 H₂O₂ → HO₂ + H⁺ + OH⁻ 2.5 x 10⁻⁵ s⁻¹ 29 H₂O₂ → H⁺ + OH⁻ 1.4 x 10⁻⁷ s⁻¹ 30 O₂ + O₂ → O₂ + HO₂ + OH⁻ 0.3 31 O₂ + H₂O₂ → O₂ + OH + OH 16 32

(2) O3 + H → O2 + OH k2 = 1.78×10^-11 cm^3 s^-1 (3) O + OH → O2 + H k3 = 4.40×10^-11 cm^3 s^-1 (5) O + HO2 → O2 + OH k5 = 3.50×10^-11 cm^3 s^-1 (6) H2O + O → 2 OH k6 = 5.40×10^-12 cm^3 s^-1 (9) OH + HO2 → O2 + H2O k9 = 4.00×10^-11 cm^3 s^-1 (10) HO2 + HO2 → O2 + H2O2 k10 = 2.50×10^-12 cm s^-1 (11) O + O2 + M → O3 + M k11 = 1.05×10^-34 cm^6 s^-1 (14) H + O2 + M → HO2 + M k14 = 8.08×10^-32 cm^6 s^-1 (15) OH + H + M → H2O + M k15 = 3.31×10^-27 cm^6 s^-1 (16) O2 + hv → 2 O k16 = (1.26×10^-8 s^-1) φ (17) H2O + hv → H + OH k17 = (3.4×10^-6 s^-1) φ (18) O3 + hv → O2 + O k18 = (7.10×10^-8 s^-1) φ