Linear Functions

    A distinguishing feature of a line is its constant inclination, or slope. Lines and linear functions are defined and graphs are sketched using various forms of the equation of a line. Then the question is reversed, when it is shown how to find the equation of a line given some geometric information. For example, find the equation of the line passing through two given points, or find the equation of the line passing through a point and perpendicular to another line.

Definition (Slope of a Line) Let and represent changes in the variables and respectively. The slope of a line is defined as and if coordinates are given for two points say, and then the slope of the line is given by the formula



whenever

    Horizontal lines have slope 0 and have equation for some real number Vertical lines do not have a slope and have equations of the form for some real number All non-vertical lines have equation where is the slope of the line and is the intercept.

Example (Slope of a Line) Given the two points and the slope of the line through these two points is Equivalently, the slope of the line is

Definition (Linear Function) Linear functions are functions that have the form where is the slope of the line and is the intercept. In the special case we also call these functions constant functions.

Example (Linear Function) We say the function is increasing (or rising) when and is decreasing (or falling) when The function is increasing and has graph:



and the function is decreasing and has graph:



Notice that the domain and range for all lines (with non-zero slope) is all real numbers.

Definition (Standard Form of an Equation of the Line) The standard form for the equation of the line is where and are real numbers and and are not both

    Every line can be put in standard form, including horizontal lines: and vertical lines:

Example (Standard Form of an Equation of the Line) A line is particularly easy to graph given in standard form. To sketch the graph of a line we merely need to plot two points. For example, given we can graph by choosing and using we have the point and by choosing and using we have the point This method is fast and easy to do and especially nice since we have also plotted the intercepts.

    Linear functions and lines are used very often and so have several named forms:

    (i) Standard:
    
    (ii) Slope Intercept:  
    
    (iii) Point-Slope:
    
    (iv) Two Intercept:
    
In the point-slope form the is the slope and the line goes through the point . In the two intercept form the points and are the intercepts.

Definition (Parallel and Perpendicular Lines) Two lines are parallel if they have the same slope and two lines are perpendicular when the product of their slopes is

Example (Parallel and Perpendicular Lines) The two lines and are parallel since both have slope of The two lines and are perpendicular since the product of the slopes and is

Example (Finding Equations of a Line) Given some geometric information find the equation of the line.

(a) Find the equation of the line passing through the two points and The equation of the line has the form so we find the slope: Thus, To find the intercept, we will use a point on the line, either or So we have and thus Therefore, the equation is

(b) Find the equation of the line that passes through the point and is parallel to the line The equation of the line has the form so we find the slope by solving   for We have and so Therefore, the slope is So the equation so far is To find we need a point that the line passes through and so we use and we have Therefore, and so the equation of the line is

(c) Find the equation of the line that passes through the point and is perpendicular to the line The equation of the line has the form and we find the slope by solving   for We have and so Therefore, the slope of the line is The slope of the line that we are looking for is So the equation so far is To find we need a point that the line passes through and so we use and we have Therefore, and so the equation of the line is

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