Pre-calculus: Straight Line Modification
What is a linear relationship (straight line)?
A linear relationship between x and y means the slope of the function is
always a constant.
Slope of a linear function, m
The slope of a linear function is the proportionality constant between Δx
and Δy. That is the fraction, Δy/Δx.
If the slope of a linear function, m, is a positive number, it means that if I
start at any point on the line and I travel 1 unit to the right (+x direction)
along the line, then I will gain m units in height.
If the slope of a linear function, m, is a negative number, it means that if I
start at any point on the line and I travel 1 unit to the right (+x direction)
along the line, then I will lose m units in height.
Example: y = 2 x + 5
Pre-calculus: Straight Line Modification
Let say my starting point is at (-2,1). Since the slope of the graph is 2, if I
travel one unit to the right along the line, I will gain 2 units in height and
reach (-1,3).
Because it is a linear relationship between x and y, this gain ratio is the
same everywhere along the line.
How to modify a straight line?
There are several ways to represent a linear relationship between two
variables. The following methods are logically equivalent:
Slope-Intercept form: y = m x + c
This is the most common way to represent a straight line because readers
can easily obtain two basis of a straight line, namely slope (m) and yintercept (c).
If a straight line has a slope of -3 and a y-intercept of 6, then the equation
of the straight line is y = - 3 + 6 .
Point-slope form:
y - y1
=m
x - x1
This is another common way to represent a straight line. To construct an
equation using this form, we need to know a point on the line (x1,y1) and the
slope of the line (m).
The idea of the point-slope form is that two point on the line (x,y) and (x1,y1)
will always have the same slope.
If we know a straight line passes through (3,7) and have a slope of 2, then
substituting x1 = 3 , y1 = 7 and m = 2 into the formula, we get:
y-7
=2
x-3
Two-point form:
y - y1 y1 − y2
=
x - x1 x1 − x2
This is a variation of the point slope form. To construct an equation using
this form, we need to know two points on the line (x1,y1) and (x2,y2).
The left side of the two-point form is exactly the same as the left side of the
point-slope form. The right side of the equation is just the slope of the line.
The approach and idea for two-point form is exactly the same as those for
the point-slope form.
Pre-calculus: Straight Line Modification
Intercept form:
x y
+ =1
a b
This is the least common way to represent a straight line. The word
intercept here refers to both the x-intercept (a,0) and the y-intercept (0,b) of
the straight line.
This is the best way to represent a straight line if you want the readers to
obtain two intercepts at the same time.
To see b is the y-intercept: set x = 0, and the first fraction will become zero.
The remaining part will immediately tell y-intercept = b.
To see a is the x-intercept: set y = 0, and the second fraction will become
zero. The remaining part will immediately tell x-intercept = a.
However, there is a limitation to this method is that this representation
cannot be constructed when either one of the intercepts is zero (math error:
divided by zero). Actually, it is only possible that both a and b are non-zero
or both a and b are zero (when x-intercept is zero, y-intercept must also be
zero too). In this case, we can just use the slope-intercept form with the
intercept be zero.