Why in the world do I think this graph is famous. For me it is because of how tricky it is for students to correctly draw this graph in a cartesian plane or figure out in which quadrant it should be when given an equation, and whether it should be a decreasing or increasing function. This complication is a result of a and b, the variable in the exponential equation simultaneously used to figure out where and how to draw the exponential function.

Th equation y=ab^(x-p)+q is of the exponential function where b is greater than zero and b cannot be equal to 1. Do not confuse the value of a and b, they are two separate numbers and can be confused for one when a is negative 1.

P and q do not disappoint, they never change. Here once again, they are the vertical and horizontal shifts respectively, and p does that thing of changing signs. This was explicitly detailed on the post of the parabola and again on that of the hyperbola. But the exponential function has only one asymptote, the horizontal asymptote denoted by the value of q.

Now the shape:

1. When a is equal to 1 or at least positive and b is greater than 1

The graph will be an increasing function from the left to the right and drawn above the asymptote

• When a=1 and b is between 0 and 1

The graph will be a decreasing function from left to right and drawn above the asymptote

• When a is negative and b is greater than 1

The graph is below the asymptote and is a decreasing function

• When a is negative and b is between 0 and 1

Once again the graph is below the asymptote, but an increasing function

We can see from the above conditions that a reflects if the graph is positive or negative, and if it is drawn above or below the asymptote. But b does not reflect anything in the four conditions. However, b greater than 1 is an increasing function and a decreasing function when b is between 0 and 1, the third and fourth conditions are just the mirror refection of the first two graphs along the asymptote.

To sketch this graph:

• Determine the asymptote and draw it on the cartesian plane
• Determine the y-intercept. And the x intercept if it is defined, in some cases we will not have the x intercept.
• Determine the relationship between a and b using the four conditions and draw the graph.

We are one step done with the concepts of functions, study the application examples on your study guide and reflect to these posts in order to keep the concepts at the back of your head. Let’s work on the next post series detailing the trigonometric functions.

Operating Disclaimer:

This blog post is not the first step towards learning the concepts of math but the coupling step to ease the use of your study guide and other study material. Read it over and over again to keep the concepts at the back of your head. Math is a subject of rules; know the rules, be able to duplicate them on a blank paper and go fetch your distinction