9 ## LINEAR FUNCTIONS## The Equation of a Straight LineWE NOW BEGIN THE STUDY OF 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 coördinate geometry consists in recognizing this relationship between equations and their graphs. Hence the student should know that the graph of any first degree polynomial Sketching the graph of a first degree equation should be a basic skill. See Lesson 33 of Algebra.
Example. Mark the
The The Now, what does it mean to say that It means that every coördinate pair ( That line, therefore, is called the graph of the equation 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 linear equations.
Problem 1. Mark the
To see the answer, pass your mouse over the colored area. The
Problem 2. Sketch the graph of An equation of the form See Lesson 33 of Algebra, the section "Vertical and horizontal lines." The slope-intercept form This linear form
is called the slope-intercept form of the equation of a straight line. Because, as we shall prove presently, This first degree form
where
Theorem.
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 Then if (
On solving for
Therefore, since the variables The slope of a straight line -- that number -- indicates the rate at which the value of Problem 3. Name the slope of each line, and state the meaning of each slope. a)
The slope is 2. This means that
c)
The slope is 1. This means that See Lesson 5. d) 3
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