**Introduction**

Graphs are diagrams of the functions and relations that express phenomena in the physical world. There are all kinds of graphs; the simplest are two-dimensional drawings. The most sophisticated graphs cannot be envisioned even by the most astute human beings, and computers are required to show cross sections of them so that we can get a glimpse of what is going on. In this chapter we will look at the most commonly used methods of graphing. There will be plenty of examples so that you can see what the graphs of various relations and functions look like.

**Rectangular Coordinates**

The most straightforward two-dimensional coordinate system is the *cartesian plane* (Fig. 3-1), also called *rectangular coordinates* or the *xy plane* . The *independent variable* is plotted along the *x* axis, or *abscissa* ; the *dependent variable* is plotted along the *y* axis, or *ordinate* . The scales of the abscissa and ordinate are usually (but not always) linear, and they are perpendicular to each other. The divisions of the abscissa need not represent the same increments as the divisions of the ordinate.

**Fig. 3-1** . The cartesian coordinate plane.

**Slope-intercept Form Of Linear Equation**

A linear equation in two variables can be rearranged from standard form to a conveniently graphable form as follows:** **

*ax* + *by* + *c* = 0

*ax* + *by* = − *c*

*by* = − *ax* − *c*

*y* = (− *a/b* ) *x* − ( *c/b* )

where *a, b* , and *c* are real-number constants, and *b* ≠ 0. Such an equation appears as a straight line when graphed on the cartesian plane. Let *dx* represent a small change in the value of *x* on such a graph; let *dy* represent the change in the value of *y* that results from this change in *x* . The ratio *dy/dx* is defined as the *slope* of the line and is commonly symbolized *m* . Let *k* represent the *y* value of the point where the line crosses the ordinate. Then the following equations hold:

*m* = − *a/b*

*k* = − *c/b*

The linear equation can be rewritten in *slope-intercept form* as

*y* = *mx* + *k*

To plot a graph of a linear equation in cartesian coordinates, proceed as follows:

- Convert the equation to slope-intercept form.
- Plot the point
*y*=*k*. - Move to the right by
*n*units on the graph. - Move upward by
*mn*units (or downward by −*mn*units). - Plot the resulting point
*y*=*mn*+*k*. - Connect the two points with a straight line.

Figures 3-2 and 3-3 illustrate the following linear equations as graphed in slope-intercept form:

*y* = 5 *x* − 3

*y* = − *x* + 2

A positive slope indicates that the graph “ramps upward” and a negative slope indicates that the graph “ramps downward” as you move toward the right. A zero slope indicates a horizontal line. The slope of a vertical line is undefined because, in the form shown here, it requires that something be divided by zero.

**Point-slope Form Of Linear Equation**

It is not always convenient to plot a graph of a line based on the y-intercept point because the part of the graph you are interested in may lie at a great distance from that point. In this situation, the *point-slope form* of a linear equation can be used. This form is based on the slope *m* of the line and the coordinates of a known point ( *x* _{0} , *y* _{0} ):

*y* − *y* _{0} = *m* ( *x* − *x* _{0} )

To plot a graph of a linear equation using the point-slope method, you can follow these steps in order:

- Convert the equation to point-slope form.
- Determine a point (
*x*_{0},*y*_{0}) by “plugging in” values. - Plot (
*x*_{0},*y*_{0}) on the plane. - Move to the right by
*n*units on the graph. - Move upward by
*mn*units (or downward by −*mn*units). - Plot the resulting point (
*x*_{1},*y*_{1}). - Connect the points (
*x*_{0},*y*_{0}) and (*x*_{1},*y*_{1}) with a straight line.

Figures 3-4 and 3-5 illustrate the following linear equations as graphed in point-slope form for regions near points that are a long way from the origin:

*y* − 104 = 3 ( *x* − 72)

*y* + 55 = −2 ( *x* + 85)

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