244 Graphs of Linear Relations
Are you confused?
The PS form of a linear function is actually a generalized version of the SI form. The PS form is handy
when we don’t know the y-intercept of a graph, but we do know the coordinates of some point in one of
the quadrants. If we are told only those coordinates and the slope, we can easily write down an equation
representing the function using the PS form. We can then draw the graph by finding another point using
that equation, and connecting the two points with a straight line.Here’s a challenge!
Look again at the general PS form for a linear equation, whose graph has a known point with coordinates
(x 0 y 0 ) and a slope m, and where x is the independent variable and y is the dependent variable:y − y 0 =m(x − x 0 )Convert this equation into SI form.Solution
Table 15-2 is an S/R derivation that shows how this can be done.Equations from Graphs
Let’s derive the SI and PS forms of linear equations by looking at how their graphs behave
generally. Then we’ll derive a standard form for a linear equation based on two known points.Known slope and y-intercept
Imagine a line in Cartesian coordinates that has slope m and crosses the y axis at the point
(0,b), as shown in Fig. 15-6. If we move away from (0, b) on the line, the slope is always equal
to the difference in the y value divided by the difference in the x value, or Δy/Δx.Table 15-2 Conversion of a linear equation from PS to SI form.
Remember that m,x 0 , and y 0 are constants. The y-intercept, called
b in the classical expression of the SI form, turns out to be the
quantity ( y 0 −mx 0 ).
Statements Reasons
y−y 0 =m(x−x 0 ) This is the equation we are given
y−y 0 =mx−mx 0 Distributive law of multiplication over subtraction
y=mx−mx 0 +y 0 Add y 0 to each side
y=mx+ (−mx 0 )+y 0 Change subtraction to negative addition
y=mx+ (−mx 0 +y 0 ) Grouping of addends
y=mx+ (y 0 −mx 0 ) Simplify second addend on right side