LINEAR FUNCTIONSThe Equation of a Straight LineSketching the graphThe slope-intercept formThe proofWE NOW BEGIN THE STUDY OF THE GRAPHSofpolynomial functions.We will find that the graph of eachdegreeleaves its characteristic signature on thex-y-plane.

The graph of a first degree polynomial is always a straight line. The graph of a second degree polynomial is a curve known as a parabola. Apolynomial of the third degree has the form shown on the right. Skill in cordinate geometry consists in recognizing this relationship between equations and their graphs. Hence the student should know that the graph of any first degree polynomialy=ax+b is a straight line, and, conversely, any straight line has for its equation,y=ax+b.Sketching the graph of a first degree equation should be a basic skill. SeeLesson 33 of Algebra.Example.Markthex- andy-intercepts, and sketch the graph ofy= 2x+ 6.Solution.

Thex-intercept is theroot. It is the solution to 2x+ 6 = 0. Thex-intercept is 3.They-intercept is theconstant term, 6.Now, what does it mean to say that y= 2x+ 6 is the "equation" of that line?It means that every cordinate pair (x,y) that is on the graph,solvesthat equation. (That's what it means for a cordinate pair to be on the graph on any equation.) Every cordinate pair (x,y) on that line is(x,2x+ 6).That line, therefore, is called thegraphof the equationy= 2x+ 6. And y= 2x+ 6 is called theequationof that line.Every first degree equation has for its graph a straight line. (We willprovethat below.) For that reason, functions or equations of the first degree -- where 1 is the highest exponent -- are calledlinearfunctions or linear equations.Problem 1.Mark thex- andy-intercepts, and sketch the graph of y= 3x 3To see the answer, pass your mouse over the colored area.To cover the answer again, click "Refresh" ("Reload").

Thex-intercept is the solution to 3x 3 = 0. It isx= 1. They-intercept is the constant term, 3.Problem 2.Sketch the graph ofy= 4.

An equation of the formy= A number, is a horizontal line.SeeLesson 33 of Algebra, the section "Vertical and horizontal lines."The slope-interceptformThis linear formy=ax+bis called theslope-intercept formof the equation of a straight line. Because, as we shall prove presently,ais theslopeof the line (Topic 8), andb-- the constant term -- is they-intercept.This first degree formAx+By+C= 0whereA,B,Careintegers, is called thegeneral formof the equation of a straight line.Theorem.Theequationy=ax+bis the equation of a straight line with slopeaandy-interceptb.For, a straight line may be specified by giving its slope and the cordinates of one point on it. (Theorem 8.3.)Therefore, let the slope of a line bea, and let the one point on it be itsy-intercept, (0,b).

Then if (x,y) are the cordinates ofanypoint on that line, itsslopeisybx 0=ybx=a.

On solving fory,y=ax+b.Therefore, since the variablesxandyare the cordinates ofanypoint on that line, that equation is the equation of a straight line with slopeaandy-interceptb. This is what we wanted to prove.The slope of a straight line -- that number -- indicates therateat which the value ofychanges with respect to the value ofx. (Topic 8.)Problem 3.Name the slope of each line, and state the meaning of each slope.a) y= 2x+ 6The slope is 2. This means thatyincreases 2 units for every 1 unit ofx.b) y= 23x + 4

The slope is 23. This means thatydecreases 2 units for every

3 units ofx.

c) y=xThe slope is 1. This means thatyincreases 1 unit for every 1 unit ofx. This is theidentity function, Lesson 5.d) 3x+ 3y= 1It is only when y=ax+b, that the slope isa. Therefore, on solving fory: y= x+ 1/3. The slope is 1. This means thatydecreases 1 unit for every unit thatxincreases.