Hello. I'm Professor Von Schmohawk and welcome to Why U.
We have seen that many functions encountered in mathematics
are real-valued functions of a real variable.
These functions take their input values from a real variable, called the "independent variable"
and for each valid input value, produce one specific real output value.
When "x" is used as the input variable
the output value is typically assigned to the variable "y"
called the "dependent variable".
These values of x and y can be visually represented on the Cartesian plane.
This visual representation is called the "graph" of the function
and the process of creating it is called "graphing" the function.
Real-valued functions of a real variable are typically described by an equation.
For example, the equation which describes the function represented by this graph
is "y equals two x".
This function is said to be "linear"
since its graph is a line in the Cartesian plane.
Linear functions are very simple but extremely important in mathematics.
This is not only because many natural phenomena can be modeled by linear equations
but also because more complicated functions
often appear linear when viewed on a small enough scale.
Notice that as we examine smaller and smaller regions of this curve
the graph looks more and more linear.
The fact that many functions appear linear on a small scale
is the logical basis of branches of mathematics such as calculus.
It is this same principle which allows us to treat the Earth as a flat plane locally
even though on a large scale its surface is actually curved.
Lines are simple geometric shapes
and so the equations which describe them are also simple.
As an example, let's create an equation which describes this linear graph.
Notice that the x and y coordinates of each point on this line have the same value.
For instance, when x has a value of five
y also has a value of five.
This equality between x and y applies to every point on this line.
Therefore, the equation "y equals x" describes this function.
This equation states the condition
that the x and y coordinates of every point on the graph have the same value.
We say that the coordinates of any point on this line "satisfy" this equation.
We can verify this by replacing the variables x and y in the equation
by the coordinates of any individual point.
If this creates a true statement
then the coordinates satisfy the equation and the point lies on the graph.
For instance, the coordinates (3,3) satisfy this equation
since replacing both x and y by three
creates the true statement "three equals three".
Therefore, the point whose coordinates are (3,3)
lies on this line.
On the other hand, we can confirm that the point (4,3) does not lie on this line
since substituting these coordinates into the equation creates a false statement.
Now instead of the equation "y equals x"
let's see what graph is created by the equation "y equals two x".
In this equation, the variable x is multiplied by the constant, "two".
A constant multiplier of a variable is called the "coefficient" of the variable.
In this equation "two" is the coefficient of the variable x.
This equation states that the y-coordinate of each point in the graph
has a value twice that of the x-coordinate.
For example, if the x-coordinate of a point is three
then the y-coordinate of that point must be six.
Or if x is two
then y will be four.
If we continue to calculate more points for this equation
we will see that these points all fall on a line running through the origin.
We can confirm that this line passes through the origin
since substituting the coordinates (0,0) into this equation, results in a true statement.
Therefore, the point (0,0) lies on this line.
Let's see what happens if we make the x-coefficient even larger
say, three.
We can see that if x equals two
y will equal three times that value, or six.
So the point (2,6) lies on this graph.
Calculating y for several more values of x
we see that once again, the collection of all the points which satisfy this equation
form a line passing through the origin.
The graphs of all of the equations which we have seen
are lines which pass through the origin and slope upwards as we move towards the right.
The only difference between these equations is the coefficient of x.
Notice that the larger the x-coefficient, the greater the slope of the line.
But what if the x-coefficient is negative?
For example, let's say we pick an x-coefficient of negative one.
Now if x is equal to three
y must be negative three.
Likewise, if x is negative three
then y will be positive three.
This equation also describes a line which passes through the origin
but slopes in the opposite direction from the lines formed by equations
with positive coefficients.
This line slopes downwards as we move to the right.
Increasing the magnitude of this negative coefficient
produces lines with a greater downward slope.
So the sign of the multiplier determines whether the line slopes up
or down
and the magnitude of the multiplier determines the magnitude of the slope.
Lines which slope up as we move to the right are said to have a positive slope
and lines which slope down are said to have a negative slope.
So what happens if we make the x coefficient zero?
Since zero times x is always zero regardless of the value of x
this equation can be written as "y equals zero".
This equation states that regardless of the value of the x-coordinate
every point's y-coordinate will be zero.
So this equation describes a horizontal line running through the origin.
This fits the relationship we have seen between the x-coefficient and the slope
since an x-coefficient of zero produces a graph with zero slope.
So for linear functions, the coefficient of the independent variable x
determines the slope of the graph.
The coefficient's sign determines the sign of the slope
and the coefficient's magnitude determines the magnitude of the slope.
The smaller the magnitude of the x-coefficient, the smaller the slope.
And when the x-coefficient is zero, the slope will be zero
and the graph is a horizontal line.
But is it possible for an equation to describe a vertical line?
We have seen that as the x-coefficient gets larger
the slope of the graph becomes closer to vertical.
However, no matter how large the coefficient becomes
the line will never be exactly vertical.
It would require an infinitely large x-coefficient to produce a vertical line.
But infinity is not a number and therefore cannot be used as a coefficient.
However, a vertical line can be described by the equation "x equals zero".
But this equation does not describe a function of x
since it associates multiple values of y with a single x value.
So with an equation of the form "y = mx"
where "m" is some constant
we can describe any non-vertical line passing through the origin.
By choosing the value of "m" we can create a line with any slope we please.
In the next lecture we will see how to create functions
whose graphs are lines which do not pass through the origin.