Linear Functions
A linear function is a function of the form f(x) = mx + n whose graph is a straight
line. When written in this form, the slope or gradient of it is m and the y-intercept is n.
Unless a domain for x is otherwise stated, the domain for linear functions will be
assumed to be all real numbers and so the lines in graphs of all linear functions extend
infinitely in both directions.
Also in linear functions with all real number domains, the range of a linear function
will cover the entire set of real numbers for y, unless the slope m = 0 and the function equals a
constant. In such cases, the range is simply the constant.
Examples:
graphs of f(x) = x+2 and f(x) = –2 x
Domain=range= 
(See also page 122)
Slopes and y-intercepts
increase ·in·y
and the graph hits the y-axis at the y-intercept
increase·in·x
Essentially, the absolute value of the slope measures the "steepness" of the line.
If the slope is positive, then the line "rises" from left to right: increasing function; if
the slope is negative, then the line "falls" from left to right: decreasing function; and if the
slope is 0, then the line is horizontal: constant function.
The slope is the ratio
Note: the simplest linear functions are those whose graphs pass through the origin f(x) = mx
Note: equations of the form f(x) = c are linear functions with slope 0: constant functions;
their graphs are straight horizontal lines, crossing through the y-axis at c.
Note: the graph of f(x) = 0 (m = 0, n = 0) is the x-axis; the the diagonals through the origin
are the graphs of the linear functions f(x) = x and f(x) = – x .
Note: the y-axis has the equation x = 0 which is not a function; because there are multiple
possible values of y for the single value of x, equations of the form x = c are not functions,
but can be considered relations; the lines of such equations have no slope, although some
would say the slope is infinity because the lines have infinite steepness.
Examples:
Graph A: a line parallel to the x-axis
Graph B: a line parallel to the y-axis
Graph C: a line through the origin at 45° to the x-axis
Graph D: a line with both x- and y-intercepts.
Note: The first thing you have to do to deal with straight-line functions is to rearrange the
equation into the standard format f(x) = mx + n.
Finding the intercepts of the graph of a linear function
Substitute the value x=0 in the equation to calculate the y-intercept. Substitute the
value y=0 in the equation to calculate the x-intercept.
Example:
suppose you are to find the intercepts of the graph given by 5x + 2y = 10
to find the x-intercept, set y = 0 and solve for x.
so the x-intercept point is (2,0)
5x  2·0  10  5x  10  x  2
to find the y-intercept, set x = 0 and solve for y.
5·0  2 y  10  2 y  10  y  2
so the y-intercept point is (0,5).
Finding the equation of a linear function given its graph
Once the slope and the y-intercept have been found write the equation with the
standard format. The slope is calculated as follows:
-Calculate the difference in values of the y-coordinates of two points of the graph.
-Divide it by the difference in the x-coordinates of the points ("rise divided by run").
Example:
Geometric interpretation of a pair of simultaneous linear equations
Plotting the graphs of two linear functions, the coordinates of the point at which both
graphs meet, are the solution of the system. (See page 18)
Quadratic functions
The graph of f(x)=ax2+bx+c is a parabola. Functions of this sort are called quadratic
functions.
Plotting graphs of quadratic functions
The value of a stretches the graph and makes the function to be convex or concave.
The ordinate of the y-intercept is found by calculating the value of f(x) when x = 0.
f(x)=x2-4x+3
Some parabolas meet the x-axis. The discriminant b2-4ac gives information about the
number of x-intercepts. The abscissas of these points can be found by algebraic methods:
factorisation of the function, completing the square or by using the formula
The value of –b/(2a) is the abscissa of the vertex. The vertex of the curve occurs at its
maximum or minimum point.
Homework
Page 236, 5 and 6
Example 8a page 236
Homework
Page 236, 8b
Examples:
12 b d, 21 b d page 243 and following
Homework
12 a c, 21 a c page 243 and following
Linear Equations and Functions
If the graphing of the equation forms a straight line, then the equation is considered a linear equation with two
variables. The following equation is a simple example of such a linear equation:
Even if the equation has only one of the two variables but is graphed on a two dimensional graph as a line, the
equation is still considered a linear equation.
If the graph of a function with a single input variable forms a straight line, then the function is considered a
linear function. Linear equations with two variables can usually be algebraically arranged to give the form of a
linear function. Consider, the following simple linear function f(x) defined as:
We can define y = f(x); therefore
where x and y are variables to be plotted in a two-dimentional Cartesian coordinate graph as shown here:
This function is equivalent to the previous example of a linear equation, y - x = 2. The arrows at each end of the
line indicate that the line extends infinitely in both directions. All linear functions of a single input variable have
or can be algebraically arranged to have the general form:
where x and y are variables, f(x) is the function of x, m is a constant called the slope of the line, and b is a
constant which is the ordinate of the y-intercept (i. e. the value of y where the function line crosses the y-axis).
The slope indicates the steepness of the line. In the previous example where y = x + 2, the slope m = 1 and the yintercept ordinate b = 2. The y = mx+b form of a linear function is called the slope-intercept form.
Unless a domain for x is otherwise stated, the domain for linear functions will be assumed to be all real numbers
and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with
all real number domains, the range of a linear function will cover the entire set of real numbers for y, unless the
slope m = 0 and the function equals a constant. In such cases, the range is simply the constant.
Conversely, if a function has the general form y = m x + b or if it can be arranged to have that form, the function
is linear. A linear equation with two variables has or can be algebraically rearranged to have the general form1:
where x and y represent the linear variables, and the letters A, B, and C can represent any real constants, either
positive or negative. Conversely, an equation with two variables x and y having that general form, or being able
to be arranged in that form, would be linear as long as A and B are not both equal to 0. In the preceding equation,
capital letters are to avoid confusion with other constants in this chapter and for consistency with Reference 1.
If one divides the preceding equation by B (when B is not 0) and solves for y, the following form can be
obtained:
If one equates -A/B to the slope m and C/B to the y-intercept ordinate b, it can be seen that the general form for a
linear equation and the slope-intercept form for a linear function are practically interconvertible except for the
fact that, in a linear function, the B constant in the linear equation form cannot equal 0.
[edit]
Slope
Two different points plotted on a plane are enough to determine the identity of a straight line, if it is assumed
these points are part of the line. So, if the coordinates of points (x 1,y1) and (x2,y2) are known, then a straight line
is defined on the graph and a two variable linear equation can be determined, unless both x values are equal and
both y values are equal. If x1 = x2 and y1 = y2, then the two points are the same and many lines can go through
the point. If x1 = x2, but
y1 ≠ y2, then the linear equation is not a function because there is more than one y value
per x value. If x1 ≠ x2, then the line going through the points defines a linear function of x.
For a linear function, the slope can be determined from any two known points on the line. The slope corresponds
to an increment or change in the vertical direction divided by a corresponding increment or change in the
horizontal direction between any different points on the straight line.
Let Δy = increment or change in the y-direction (vertical) and
Let Δx = increment or change in the x-direction (horizontal).
For two points (x1,y1) and (x2,y2), the slope of the function line m is given by:
 This formula is called the
formula for slope measure but
is sometimes referred to as the
slope formula.
For a linear function, knowing any two different points on the line or knowing the slope and any one point on the
line is enough to determine the line and identify it by an equation. There is an equation form for a linear function
called the point-slope form of a line2 which uses the slope m and any one point (x1,y1) to determine a valid
equation for the function's line:
Example: Find the slope and function of the line connecting the points (2,1) and (4,4).
Solution: When calculating the slope of a straight line from two points with the preceding formula, it does not
matter which is point 1 and which is point 2. Let's set (x1,y1) as (2,1) and (x2,y2) as (4,4). Then using the twopoint formula for the slope m:
Using the point-slope form:
One substitutes the coordinates for either point into the point-slope form as x1 and y1. For simplicity, we will use
x1=2 and y1=1.
Using the slope-intercept form:
Alternatively, one can solve for b, the y-intercept ordinate, in the general form of a linear function of one
variable, y = m x + b.
Knowing the slope m, take any known point on the line and substitute the point coordinates and m into this form
of a linear function and calculate b. In this example, (x1,y1) is used.
Now the constants m and b are both known and the function is written as
or alternatively as
_________________________________________end of example_____________________________________
For completely horizontal lines, the difference in y coordinates between any two points is 0, so the slope m = 0,
indicating no steepness in the line at all. If the line extends between right-upper (+,+) and left-lower ( -, -)
directions, then the slope is positive. As the slope increases, the line become steeper until the line is almost
vertical when the slope is very large. When the slope m = 1, the line is diagonal with an angle halfway between
the x and y axes. If the line extends between left-upper (-,+) and right-lower (+, -) directions, then the slope is
negative. As the slope changes from 0 to very negative numbers, the steepness in the opposite direction
increases. Compare the slope ( m ) values in the following graph of functions y = 1 (where
m = 0), y = (1/2) x + 1, y = x + 1, y = 2 x, y = -(1/2) x + 1, y = -x + 1, and y = -2 x + 1. For all two-variable linear
equations that can be converted to linear functions, the same calculation applies to slopes for those lines.
[edit]
x=c and y=c equation lines, where c = constant
This section discusses the graphing of simple linear equations of the form x = c and y = c, where x and y are
variables and c is any real number constant.
In the equation x = c, x can only be the real number c, and since nothing is said about y, y can be equal to any
real number regardless of x. Therefore, the graph of x = c is a straight vertical line where x = c, but covering all
positive and negative values of y (see the following diagram).
The domain of this relation is just the set { c } and the range covers all real numbers (unless otherwise specified).
Because there are multiple possible values of y for the single value of x, equations of the form x = c are not
functions, but can be considered relations. The lines of such equations have no slope, although some would say
the slope is infinity because the lines has infinite steepness. Other lines where the steepness approaches vertical
have very large slopes. These are the only types of linear equations of the general form shown previously which
are not linear functions. The equation x = 0 is equivalent to the y-axis.
Equations of the form y = c are linear functions of the general form y = m x + b where m = 0 and the constant c
equals the y-intercept b (in the general form). Since the slope is 0, the graph of this function is a straight
horizontal line, crossing through the y-axis at c, and extends infinitely in the positive and negative directions for
all values of x (see the following diagram).
The domain for such functions covers all real numbers (unless otherwise specified), but the range is just the set {
c }. The equation y = 0 is equivalent to the x-axis.
[edit]
x and y axes intercepts
An axis intercept point is a point where the graph of a function, relation, or equation intersects one of the axes.
Alternatively, an axis intercept may simply refer to the number value on the axis where the intersection occurs.
An axis intercept on the x-axis is called an x-intercept.
If the intersection with the x-axis occurs at (a,0), then the x-intercept point is (a,0) and the
x-intercept is often simply referred to as a. Likewise, an axis intercept on the y-axis is called a
y-intercept. If the intersection with the y-axis occurs at (0,b), then the y-intercept point is (0,b) and the yintercept is often simply referred to as b. For lines based on equations of the form
x = c, the x-intercept is the point (c,0) and there is no y-intercept unless x = 0. For lines based on equations of the
form y = c, the y-intercept is the point (0,c) and there is no x-intercept unless y = 0. For linear equations or
functions that can be expressed in the form
where the slope m is any constant not equal to 0, the x-intercept and y-intercept are the same point, (0,0), with no
other axis intercepts. For all other linear equations with two variables, x and y, the lines extending infinitely in
both directions have to cross both axes somewhere; therefore, such equations will have one x-intercept and one
y-intercept. For that sort of equation, knowing where both intercepts are is enough to be able to determine and
draw the line through the points to graph the equation.
When one is given a function or equation of two variables which is apparently linear and the goal is to graph the
line, finding the axes intercepts is often the easiest way to go about doing it. To find the x-intercept, set y = 0 and
solve for x. To find the y-intercept, set x = 0 and solve for y.
If the intercepts are different points, then a line can be drawn through the two intercept points.
If both intercepts are (0,0), then another point must be determined to graph the line from two points or the slope
can be determined and used to plot the line going though the origin (0,0).
Of course if the equations are of the form x = c or y = c, the horizontal or vertical lines are very simple to plot.
Also, after the intercepts are found, the slope can be calculated from two different intercept points using the twopoint slope formula previously mentioned.
Example: Graph the equation 5x + 2y = 10 and calculate the slope.
Solution: This fits the general form of a linear equation, so finding two different points are enough to determine
the line. To find the x-intercept, set y = 0 and solve for x.
so the x-intercept point is (2,0). To find the y-intercept, set x = 0 and solve for y.
so the y-intercept point is (0,5). Drawing a line through (2,0) and (0,5) would produce the following graph.
Graph of 5x + 2y = 10 showing intercepts
To determine the slope m from the two points, one can set (x1,y1) as (2,0) and (x2,y2) as (0,5), or vice versa and
calculate as follows:
_________________________________________end of example_____________________________________
[edit]
Intercept Form of a Line
There is one more general form of a linear function we will cover. This is the intercept form of a line, where the
constants a and b are such that (a,0) is the x-intercept point and (0,b) is the
y-intercept point.
where a ≠ 0
and b ≠ 0
Neither constant a nor b can equal 0 because division by 0 is not allowed. The intercept form of a line cannot be
applied when the linear function has the simplified form y = m x because the
y-intercept ordinate cannot equal 0.
Multiplying the intercept form of a line by the constants a and b will give
which then becomes equivalent to the general linear equation form A x + B y + C where A = b, B = a, and C =
ab. We now see that neither A nor B can be 0, therefore the intercept form cannot represent horizontal or vertical
lines. Multiplying the intercept form of a line by just b gives
which can be rearranged to
which becomes equivalent to the slope-intercept form where the slope m = -b/a.
Example: A graphed line crosses the x-axis at -3 and crosses the y-axis at -6. What equation can represent this
line? What is the slope?
Solution: intercept form:
Multiplying by -6 gives
so we see the slope m = -2.
Graph of y = - 2x - 6 showing intercepts.
The line can also be written as
Example: Can the equation
be transformed into an intercept form of a line, (x/a) + (y/b) =1, to find the intercepts?
Solution: No, no amount of valid mathematical manipulation can transform it into the intercept form. Instead
multiplying by 4, then subtracting 2x gives
which is of the form y = m x where m = -2. The line intersects the axes at (0,0). Since the intercepts are both 0,
the general intercept form of a line cannot be used.
_________________________________________end of example_____________________________________
[edit]
Summary of General Equation Forms of a Line
The most general form applicable to all lines on a two-dimensional Cartesian graph is
with three constants, A, B, and C. These constants are not unique to the line because multiplying the whole
equation by a constant factor gives a new set of valid constants for the same line. When B = 0, the rest of the
equation represents for a vertical line, which is not a function. If B ≠ 0, then the line is a function. Such a linear
function can be represented by the slope-intercept form which has two constants.
slope-intercept form:
The two constants, m and b, used together are unique to the line. In other words, a certain line can have only one
pair of values for m and b in this form.
The point-slope form given here
uses three constants; m is unique for a given line; x1 and y1 are not unique and can be from any point on the line.
The point-slope cannot represent a vertical line.
The intercept form of a line, given here,
a ≠ 0 and b
≠0
uses two unique constants which are the x and y intercepts, but cannot be made to represent horizontal or vertical
lines or lines crossing through (0,0). It is the least applicable of the general forms in this summary.
Of the last three general forms of a linear function, the slope-intercept form is the most useful because it uses
only constants unique to a given line and can represent any linear function. All of the problems in this book and
in mathematics in general can be solved without using the point-slope form or the intercept form unless they are
specifically called for in a problem. Generally, problems involving linear functions can be solved using the
slope-intercept form
(y = m x + b) and the formula for slope.
[edit]
Discontinuity in otherwise Linear Equations
Let's look at the function
.
If we plot points and graph this function, we see that we get a straight line except for a point at
x = 1. At x = 1, the denominator, x - 1, becomes 0, and the function is not defined at this point.
The otherwise linear form of this function can be explained by the fact that the expression
2 x2 - 5 x + 3 in function y can be factored as follows:
for all x except 1
The factor (x - 1) can be canceled out from the numerator and denominator for all x except
x = 1, where x is undefined. The remaining expression, 2 x - 3, is a linear function of x, with a slope m = 2 and a
y-intercept ordinate of -3. The expression, 2x - 3, would be equal -1 at x = 1, but function y is not defined at that
point because that would make the denominator = 0. Since the function is not defined at x = 1, we say there is a
discontinuity for the function y at x = 1, practically speaking a sort of one-point hole in the function, shown on
the graph as a small hollow circle at that point.
Example: What would the graph of the following function look like?
Solution:
The (x + 2) factors cancel in both numerator and denominator. This makes y = x - 2 for all x except x = -2, where
there is a discontinuity. The line y = x - 2 would have a slope m = 1 and a
y-intercept ordinate of -2. So for the final answer, we graph a line with a slope of 1 and a y-intercept of -2, and
we show a discontinuity at x = -2, where y would otherwise have been equal to -4.
Example: Write a function which would be graphed as a line the same as y = 2 x - 3 except with two
discontinuities, one at x = 0 and another at x = 1.
Solution: We need the following two factors in the denominator to make it 0 at both those values of x:
denominator = (x - 0)(x - 1) = x (x - 1)
We also multiply the numerator by this expression in the function in order to cancel it out in the denominator,
then we can multiply the expressions out as shown: