We begin our serious look at functions with linear functions , whose graphs are
lines . Lines

are ubiquitous in calculus: in fact, differential calculus is primarily focused
in using lines (linear

functions) to approximate curves (nonlinear functions). Consequently, they
warrant some special

attention:

**Definition:** A vertical line is a set of all points satisfying the
equation x = c for some fixed

c. Let f(x) = mx + b be a linear function. We call the graph of f a
(**non-vertical**) line in the

plane. We call m the slope of the line , and b the **y-intercept.**

**Example:** Given two points in the plane P_{1}(x_{1}; y_{1}) and P_{2}(x_{2}, y_{2}), find an
equation of the

line that passes through both.

** Solution :** If x_{1} = x_{2}, then the vertical line x = x_{1} passes through them both, so
we suppose

x_{1} ≠ x_{2}. Then the line is the graph of f(x) = mx + b for some m,b,
so we need to find m and

b with the conditions that f(x_{1}) = y_{1} and f(x_{2}) = y_{2} (so that P_{1} and P_{2} are in
the graph of f).

That gives us two linear equations,

mx_{1} + b = y_{1}, and mx_{2} + b = y_{2
}

in two unknowns (m, b). From the first equation, we get b = y_{1} - mx_{1}, and
substituting this into

the second equation, we get

and since there is no m or b on the right-hand side, we have a definite value
for m . Using our

equation for b in terms of m , we see that

Therefore

and since our line is the graph, its equation is therefore y = f(x), or

From the example, we get two very important formulas: first, that the slope
of a line through

any two points P_{1}(x_{1}, y_{1}) and P_{2}(x_{2}, y_{2}) with x_{1}
≠ x_{2} is

The slope is a measurement of the steepness that a function increases or
decreases, which can be

seen in the angles the graphed lines make with the x-axis. Moreover, we see that
the slope of a

vertical line would be undefined, since we would have to divide by 0.

The second formula we ¯find is an equation for a line provided you know one point
P1(x_{1}, y_{1}),

and the slope:

y = mx + y_{1} - mx_{1},

or

y - y_{1} = m(x - x_{1}),

the **point-slope** form of a line. The more standard form of writing the equation,
y = mx+b, is

called the slope- intercept form .

**Example:** Find the equation of a line passing through the point (1, 1) with slope
7.

Solution: We're given the slope, m = 7, and an initial point (x_{1}, y_{1}) = (1, 1),
so the form

of the equation to use is point-slope:

y - y_{0} = m(x - x_{0});

or

y - 1 = 7(x - 1):

Rewriting into slope intercept form by solving for y , we get

y = 7(x - 1) + 1 = 7x - 7 + 1 = 7x - 6;

so the line we want is y = 7x - 6:

Sketch of y = 7x - 6.

As was mentioned earlier, functions give us a slightly more general way of
looking at equations:

De¯nition: We de¯ne the graph of an equation in x and y to be the set of all
points

(a, b) that satisfy the equation when a is substituted for x and b is
substituted for y. From this

de¯nition, we see that the graph of a function f(x) is really the graph of the
equation y = f(x).

So, when we say the graph of f(x) = mx + b, we also mean the graph of the
equation

y = mx + b. The two notations (f(x) and y) will be used interchangeably.

**Definition:** We say two (non-vertical) lines are parallel if they have the same
slope. We

say two lines are perpendicular if their slopes are negative reciprocals , i.e.
the product of their

slopes is -1.

**Example:** The lines y = x and y = x + 3 are parallel: both have slope 1.

Sketch of y = x and y = x + 3.

**Example:** The lines and y = -2x are
perpendicular: the slope of is
, the

slope of y = -2x is -2, and .

Sketch of y = (1/2)x and y = -2x.

Let's look now at taking a function whose graph we know, say y = f(x), and
determining

what function g(x) gives us the same shape, but shifted up/down and left/right.