Lesson 20: Graphing on a Coordinate Plane

New Concept

Welcome to Algebra! At its heart, algebra is the study of relationships. The core idea for this entire course is understanding how quantities relate to each other using variables.

When there is a relationship between two variable quantities, one variable is the independent variable and the other is the dependent variable.

Independent variable: The variable whose value can be chosen. Also called the input variable.

Property

A coordinate plane is a grid formed by a horizontal x-axis and a vertical y-axis that meet at the origin $(0,0)$. Points are located using ordered pairs $(x,y)$, where the x-coordinate is the horizontal position and the y-coordinate is the vertical position.

Examples

To plot $(5, 3)$, start at the origin $(0,0)$, move 5 units right, then 3 units up.
To plot $(-2, 1)$, start at the origin, move 2 units left, then 1 unit up.
To plot $(-4, -6)$, start at the origin, move 4 units left, then 6 units down.

Explanation

Think of it as a treasure map where 'X' marks the spot at the origin! The first number in an ordered pair, $(x,y)$, tells you how many steps to take left or right. The second number tells you how many steps to go up or down to find your point!

Property

An independent variable is the input whose value you can choose. A dependent variable is the output whose value is determined by the independent variable. The dependent variable's value always depends on the value chosen for the independent variable.

Examples

The cost of a phone bill (dependent) is determined by the amount of data used (independent).
Your final grade (dependent) is determined by the number of assignments you complete (independent).
The total cost (dependent) is determined by the number of movie tickets you buy (independent).

Explanation

It’s a cause-and-effect relationship! The number of hours you play video games (independent) affects your homework completion (dependent). You choose the game time, and that choice determines the outcome of your homework. Your choice is the cause, and the result is the effect.

Property

A solution to an equation with two variables is an ordered pair $(x, y)$ that makes the equation true. Solutions can be found by substituting values for the independent variable, $x$, to find the corresponding value of the dependent variable, $y$.

Examples

For the equation $y = 3x - 1$, if $x=2$, then $y = 3(2) - 1 = 5$. The solution is $(2, 5)$.
For the equation $y = 3x - 1$, if $x=0$, then $y = 3(0) - 1 = -1$. The solution is $(0, -1)$.
For the equation $y = 3x - 1$, if $x=-3$, then $y = 3(-3) - 1 = -10$. The solution is $(-3, -10)$.

Explanation

Think of an equation like $y = 3x - 1$ as a magic function machine. You put a number for $x$ in, the machine does its magic (multiplies by 3, subtracts 1), and spits out the corresponding $y$. The input and output pair is a perfect 'solution'!