Topics in

P R E C A L C U L U S

Table of Contents | Home

9

LINEAR FUNCTIONS

The Equation of a Straight Line

Solving a first degree equation


WE NOW BEGIN THE STUDY OF THE GRAPHS of polynomial functions. We will find that the graph of each degree leaves its characteristic signature on the x- y-plane.

The graphs of polynomial functions

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.  A polynomial of the third degree has the form shown on the right.  Skill in analytic geometry consists in recognizing this relationship between equations and their graphs -- hence the student will know that the graph of any first degree polynomial is a straight line, and, conversely, any straight line has for its equation, y =ax + b.

Let us begin with the polynomial function of the first degree:

y = ax + b.

To be specific, let us consider

y = 2x + 4.

A solution to that equation is the ordered pair (1, 6).  Because when x = 1 and y = 6, the equation is true:

6 = 2· 1 + 4

Another solution is (0, 4).  Because when x = 0 and y = 4, then

4 = 2· 0 + 4,

which again is true.

To find solutions, simply let x have any value you please -- the equation then determines the value of y.  

Problem 1.   Find three solutions to the first degree equation  y = x − 5.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

For example, (0, −5), (1, −4), (−1, −6).

Problem 2.   Which of the following ordered pairs solve the equation

y = 2x + 1 ?

(0, −1)      (0, 1)       (1, 2)       (1, 3)

(0, 1) and (1, 3)

Now, since there are two variables x and y, is it possible, on the x-y plane, to draw a "picture" of all the solutions to that equation?

First, let us list a few solutions:

y = 2x + 1

(−1, −1),  (0, 1),  (1, 3),  (2, 5)

And now let us plot these as points on the plane:

We see that all these solutions lie on a straight line.  In fact, the x, y coördinates of every point on that line will solve the equation!

Every coördinate pair (x, y) is

(x, 2x + 1).

y = 2x + 1.

That line, therefore, is called the graph of the equation y = 2x + 1.  And  y = 2x + 1  is called the equation of that line.

Every first degree equation has for its graph a straight line.  (We will prove that below.)  For that reason, functions, or equations, of the first degree -- where 1 is the highest exponent -- are called linear functions or equations.

The x- and y-intercepts

The x-intercept of the graph is the solution to the polynomial equation,  ax + b = 0.

The y-intercept is the value of y when x = 0.  The y-intercept is the constant term, b.

Example 1.   Mark the x- and y-intercepts, and sketch the graph of

y = 2x + 6.

 Solution.

y = 2x + 6

The x-intercept is the root.  It is the solution to 2x + 6 = 0.  The x-intercept is −3.

The y-intercept is the constant term, 6.

Again, what does it mean to say that  y = 2x + 6  is the "equation" of that line?

It means that at every point (x, y) on that line, the y-coördinate is 2 times the x-coördinate, plus 6.  Every coördinate pair on that line is (x, 2x + 6).

Problem 3.   Mark the x- and y-intercepts, and sketch the graph of  

y = −3x − 3

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

y = -3x - 3

The x-intercept is the solution to −3x − 3 = 0.  It is x = −1.  The y-intercept is the constant term, −3.


The slope-intercept form

This linear form

y = ax + b

is called the slope-intercept form of the equation of a straight line. Because, as we shall prove presently, a is the slope of the line, and b -- the constant term -- is the y-intercept.

This first degree form

Ax + By + C = 0

where A, B, C are integers, is called the general form of the equation of a straight line.


Theorem.   The equation

y = ax + b

is the equation of a straight line with slope a and y-intercept b.

For, a straight line may be specified by giving its slope and the coördinates of one point on it.  (Theorem 8.3.)

Therefore, let the slope of a line be a, and let the one point on it be its y-intercept, (0, b).

The slope intercept form

Then if (x, y) are the coördinates of any point on that line, its slope is

 yb
x − 0
  =   yb
  x
  =  a.

On solving for y,

y = ax + b.

Therefore, since the variables x and y are the coördinates of any point on that line, that equation is the equation of a straight line with slope a and y-intercept b.  This is what we wanted to prove.

Problem 4.   Name the slope of each line, and state the meaning of each slope.

a)  y = 2x + 1

The slope is 2.  This means that for every unit that x increases, y increases 2 units.  Over 1, and up 2.

  b)  y = − 2
3
x  + 4
The slope is − 2
3
.  This means that for every 3 units x increases,
y decreases 2 units.  Over 3 and down 2.

c)  y = x

The slope is 1.  This means that for every 1 unit that x increases, y increases 1 unit.  This is the identity function, Lesson 5.

d)  3x + 3y = 1

It is only when  y = ax + b, that the slope is a. Therefore, on solving for y:  y = −x + 1/3.  The slope is −1.  This means that for every unit the line goes over, it goes down 1.


Next Topic:  Quadratics:  Polynomials of the 2nd degree


Table of Contents | Home


Please make a donation to keep TheMathPage online.
Even $1 will help.


Copyright © 2001-2007 Lawrence Spector

Questions or comments?

E-mail:  themathpage@nyc.rr.com