The Natural Exponential Function

In this section we will introduce the natural base and define and graph the natural exponential function.

YouTube playlist for steps covering exponential functions and their graphs: 

This number is so special, just as is pi, that it has its own button on your scientific calculator. Try finding the e^x button on your calculator and using it to calculate e.

Evaluate.

Graph the following exponential functions. State the domain, range, and label the horizontal asymptote.

 

 

 

YouTube Videos:

Graphing Basic Functions

The graphs of the following basic functions can be determined by plotting points. That is, choose some x-values and then substitute them in to find the corresponding y-values. The more points you plot the better the picture. [ Free Printable Graph Paper ]

Identity Function: [ https://www.desmos.com/calculator/hbtql3gmj4 ]

Graph f(x) = x and state the domain and range.

  

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Squaring Function: [ https://www.desmos.com/calculator/rslqgtrovn ]

Graph f(x) = x^2 and state the domain and range.

  

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Cubing Function: [ https://www.desmos.com/calculator/g0uxyuv08i ]

Graph f(x) = x^3 and state the domain and range.

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Absolute Value Function: [ https://www.desmos.com/calculator/au6i2qvsyf ]

Graph f(x) = | x | and state the domain and range.

  

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Square Root Function: [ https://www.desmos.com/calculator/hwrrzgp3x2 ]

Graph f(x) = sqrt(x) and state the domain and range.

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Reciprocal Function: [ https://www.desmos.com/calculator/wl1xlsmgfs ]

Graph f(x) = 1/x and state the domain and range.

 

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Graph by Plotting Points

Graphing lines can be done in a number of ways.  This section describes a method, plotting points, that always works and can be used for many other types of equations. Notice that in a linear equation with two variables, y = 3x - 2, the y-value depends on what the x-value is. Since x is independent here, choose any real number, say x = 4, and you can find the corresponding y-value by evaluating y = 3(4) – 2 = 12 – 2 = 10. Therefore, the ordered pair (4, 10) is a point on the graph of the equation.

[ Free Printable Graph Paper ]

Example: Graph y = −2x + 6 by plotting five points.

   Step 1: Choose any five x-values.
   Step 2: Evaluate to find the corresponding y-values.
   Step 3: Plot the points and use a straight edge to draw a line through them.

   

When choosing x-values it is wise to pick some negative numbers as well as zero.  Try to find the points where the line crosses the x and y axes.  These special points are called the x- and y-intercepts.

Is the given point a solution?

Remember that we are less likely to make a mistake if we insert parentheses where we see a variable and then substitute in the appropriate values.

Find the corresponding value.

Graph by plotting points.

Example: Graph y = 2x − 4 by plotting five points.

           

Example: Graph y = −x − 2 by plotting five points.

           

Choosing a scale when creating a blank coordinate system will take some thought.  Keep in mind that the scale on the x-axis need not be the same as the scale on the y-axis.

Example: Graph y = 1/2 x − 6 by plotting five points.

           

When the coefficient of x is a fraction, choose x-values to be multiples of the denominator so that you might avoid unnecessarily tedious calculations.

Example: Graph y = −3/2 x + 6 by plotting five points.

          

Example: Graph 2x − 3y = 6 by plotting five points.

          

When dividing a binomial by a number you must divide both terms by that number.  For example, treat the −3 as a common denominator as in the previous problem.

A common error is to just divide the 6 by −3.

Video Examples on YouTube: