Research Paper Graph On A Coordinate Plane

As you remember from pre-algebra a coordinate plane is a two-dimensional number line where the vertical line is called the y-axis and the horizontal is called the x-axis. These lines are perpendicular and intersect at their zero points. This point is called the origin. The axes divide the plane into four quadrants.

A point in a coordinate plane is named by its ordered pair of the form of (x, y). The first number corresponds to the x-coordinates and the second to the y-coordinate.

The completeness property for points in the plane tells us two things

  1. Exactly one point in the plane is named given the numbers of the ordered pair and
  2. Exactly one ordered pair of numbers names a given point in the plane.

To graph a point one draws a dot at the coordinates that corresponds to the ordered pair. It's always a god idea to start at the origin. The x-coordinate tells you have many steps you have to take to the right (positive) or left (negative). And the y-coordinate tells you have many steps to move up (positive) or down (negative).

To find out the coordinates of a point in the coordinate system you do the opposite. Begin at the point and follow a vertical line either up or down to the x-axis. There is your x-coordinate. And then do the same but following a horizontal line to find the y-coordinate.

A relation is a set of ordered pairs. The first coordinate (usually the x-coordinate) is called the domain and the second (usually the y-coordinate) is called the range. If you remember from earlier chapters the domain contains values that correspond to the independent variable whereas the range contains values corresponding to the dependent variable.


Video lesson

Draw the following ordered pairs in the coordinate plane

(0, 0); (0,4); (4, -2); (-2, -4); (1, 3)

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A linear equation is an equation with two variables whose graph is a line. The graph of the linear equation is a set of points in the coordinate plane that all are solutions to the equation. If all variables represent real numbers one can graph the equation by plotting enough points to recognize a pattern and then connect the points to include all points.

If you want to graph a linear equation you have to have at least two points, but it's usually a good idea to use more than two points. When choosing your points try to include both positive and negative values as well as zero.


Example

Graft the function y = x + 2

Begin by choosing a couple of values for x e.g. -2, -1, 0, 1 and 2 and calculate the corresponding y values.

XY = x + 2Ordered pair
-2-2 + 2 = 0(-2, 0)
-1-1 + 2 = 1(-1, 1)
00 + 2 = 2(0, 2)
11 + 2 = 3(1, 3)
22 + 2 = 4(2, 4)

Now you can just plot the five ordered pairs in the coordinate plane

At the moment this is an example of a discrete function. A discrete function consists of isolated points.

By drawing a line through all points and while extending the line in both directions we get the opposite of a discrete function, a continuous function, which has an unbroken graph.

If you only want to use two points to determine your line you can use the two points where the graph crosses the axes. The point in which the graph crosses the x-axis is called the x-intercept and the point in which the graph crosses the y-axis is called the y-intercept. The x-intercept is found by finding the value of x when y = 0, (x, 0), and the y-intercept is found by finding the value of y when x = 0, (0, y).

The standard form of a linear equation is

$$Ax+By = C,\: \: A,B\neq 0$$

Before you can graph a linear equation in its standard form you first have to solve the equation for y.

$$2y-4x=8$$

$$2y-4x\, {\color{green} {+\, 4x}}=8\, {\color{green} {+\, 4x}}$$

$$2y=4x+8$$

$$\frac{2y}{{\color{green} 2}}=\frac{4x}{{\color{green} 2}}+\frac{8}{{\color{green} 2}}$$

$$y=2x+4$$

From here you can graph the equation as we did in the example above.

The graph of y = a is a horizontal line where the line passes through the point (0, a)

Whereas the graph of x = a is a vertical line that passes through the point (a, 0)


Video lesson

Draw the graph of the linear equation y =3x - 2

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