Math 1.5 Rectangular Coordinates and Graphs of Equations

Quadrant I

*Quadrant I* is the upper right of the graph and that means: x > 0, y > 0

Intercept Graphs help visualize a relationship between two variables or quantities. See pg 99, example 7 in text.

Intercept graphs have a vertical line that has a perpendicular horizontal line.

Definition: equation in two variables

An *equation in two variables,* say x and y, is a statement in which the algebraic expressions involving x and y are equal. The expressions are called *sides* of the equation.

Determining whether a point is on the graph of an equation

Are (2,0) points on the graph (do they satisfy the equation? 3x - y = 6 3(2) - 0 ?/= 6 6 - 0 = 6 True What about (1, -2) 3(1) - 2 ?/= 6 3 - 2 ≠ 6 False

Graph an equation by plotting points Graph the equation y = - 2x + 4 by plotting points

Assign values to x and solve for y to get (x,y) points x y = - 2x + 4 (x,y) -3 y = -2(-3) + 4 = 10 (-3,10) 1 y = -2(1) + 4 = 2 (1,2)

The Solution of an Equation must be...

The solution of a linear equation must be a single value of the variable, the empty set, or all real numbers.

Definition: graph of an equation in two variables

The *graph of an equation in two variables* x and y is the set of all points whose coordinates, (x,y), in the xy-plane satisfy the equation.

Definition: intercepts, x-intercept and y-intercept

The *intercepts* are the coordinates of the points, if any, where a graph crossed or touches the coordinate axes. An *x-intercept* is the coordinates of a point where the graph crosses or touches the x-axis. A *y-intercept* is the coordinates of a point where the graph crosses or touches the y-axis.

In Words graph of an equation in two variables

The graph of an equation in two variables is a geometric way of representing the set of all points that make the equation a true statement.

Quadrant II

*Quadrant II* is the upper left of the graph and that means: x < 0, y < 0

Quadrant III

*Quadrant III* is the lower left of the graph and that means: x < 0, y > 0

Quadrant IV

*Quadrant IV* is the lower right of the graph and that means: x > 0, and y < 0

Rectangular or Cartesian coordinate system

*Rectangular or Cartesian coordinate system*: The plane formed by the x-axis (runs horizontally) and the y-axis is often called the *xy-plane*, and the x-axis and y-axis are called the *coordinate axes.*

Work Smart Choosing values of x in an equation to find points

Work Smart: Experience will play a huge role in determining which x-values to choose in creating a table of values. For the time being, stare by choosing values of x around x = 0.

True or False: The graph of an equation in two variables x and y is the set of all points whose coordinates, (x,y) in the xy-plane satisfy the equation.

True: The graph of an equation in two variables x and y is the set of all points whose coordinates, (x,y) in the xy-plane satisfy the equation.

Ordered pair (xy)

When an ordered pair is given (3,4), the 3 is the x number and the 4 is the y number in the ordered pair. Always written in the (x,y) form.