2. Graphs of Linear Functions

It is very important to know how to quickly sketch straight lines. When we use math to model real-world problems, it is worthwhile to have a sense of how straight lines "work" and what they look like.

We met this topic before in the Plane Analytical Geometry section. The following section serves as a reminder for you...


a. Slope-Intercept Form of a Straight Line: y = mx + c

If the slope (also known as gradient) of a line is m, and the $y$-intercept is c, then the equation of the line is written:

y = mx + c


Example:

The line y = 2x + 6 has slope m = 2 and y-intercept c = 6.


A2_GraphLinEq__11.png

b. Intercept Form of a Straight Line: ax + by = c

Often a straight line is written in the form ax + by = c. One way we can sketch this is by finding the x- and y-intercepts and then joining those intercepts.

Example:

Sketch the line 3x + 2y = 6.


Answer


Slope of a Line

The slope (or gradient) of a straight line is given by:

MATH

We can also write the slope of the straight line passing through the points (x1, y1) and (x2, y2) as:

MATH

Using this expression for slope, we can derive the following.

 

c. Point-slope Form of a Straight Line: y − y1 = m(x − x1)

If a line passes through the point (x1, y1) and has slope m, then the equation of the line is given by:

y − y1 = m(x − x1)

 

Example:

Find the equation of the line with slope - 3, and which passes through (2, -4).


Answer





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