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The simplest functions are lineal functions. Their formulas are polynomials with degree one or cero (this is the case when the function is the constant function). Their graphs are straight lines.

A formula for these lineal functions is:

We usually call it the slope-intercept form, where b is the y-intercept and m is the slope.

The slope, also called the gradient of the line, measures the degree of inclination of the line with the horizontal, the x-axis. The slope of a line characterizes the general direction in which a line points.

The y-intercept of a function is the point at which the graph of the function crosses the y-axis. (the x value equals 0 ). The y-intercept of a linear function is:

Polynomials functions. Linear function: y-intercept | matematicasVisuales

When b is positive, the line crosses the y-axis somewhere above the x-axis (y=0) and if b is negative, the line crosses the y-axis below the x-axis.

A line is completely determined by any two distinct points along it. If (x0, y0) and (x1, y1) are two points (x0 not equal x1) on a given line we can calculate the slope of the line:

Polynomials functions. Linear function: The slope | matematicasVisuales

Notice that when the slope, m, is positive, the line slants upward to the right. The more positive m is, the steeper the line will slant upward to the right.

When the slope is negative, the line slants downward to the right, and, as the slope becomes more and more negative, the line will slant downward steeper and steeper to the right.

This is an example of a linear function with a negative slope:

Polynomials functions. Linear function: linear function with a negative slope | matematicasVisuales

Horizontal lines have slope=0. All horizontal lines are of the form y = k (where k is some number). As a function, we call it a constant function.

Vertical lines have no slope. They are not function at all. Note that if we try to calculate the slope we end up trying to divide by 0.

In applications, if we know that the functional relation between the variables is characterized by a constant rate of change, then the underlying function is linear and its slope measures this rate of change. This information can be exploited to write down a formula for the function.

The line thorugh P(x0, y0) with slope m is given by the point-slope form:

We can rearrange to write it as a function:

We can write the equation of the line though two points:

That has the advantage that we do not need to divide. We can arrive at the general form (this algebraic form has an important advantage because it include vertical lines x = k, where k is some real number):

The x-intercept of a linear function is the point where the graph of the line crosses the x-axis. A non-horizontal line always have one x-intercept. To find the x-intercept we need to solve the equation

This solution of the equation is called a root of the polynomial. In general, a function can have several x-intercepts.

Then the x-intercept of a linear function (with m not equal to 0) is:

Polynomials functions. Linear function: x-intercept, roots | matematicasVisuales

The simplest example of a Lagrange Polynomial is, of course, the linear function through two points. We can write this linear function as a simple example of a Lagrange Polynomial:

REFERENCES

Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.

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