Online College Courses for Credit

3 Tutorials that teach Graph of an Exponential Equation
Take your pick:
Graph of an Exponential Equation

Graph of an Exponential Equation

Author: Sophia Tutorial

Graph of an Exponential Equation

See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

Begin Free Trial
No credit card required

29 Sophia partners guarantee credit transfer.

312 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 27 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.


What's Covered

  • The Graph of y = bx
  • The y-intercept of y = abx
  • Other Features of the Graph

Graph of an Exponential Equation

The Graph of y = bx

Below is a sketch of the graph y = 2x  Notice that the base to the exponential expression is 2, and there is an implied scalar multiplier of 1 in front of this expression:

We see that, in this case, as x gets larger (approaches positive infinity), the value of y also increases (it approaches positive infinity as well).  On the other hand, as x gets smaller (approaches negative infinity), the value of y gets smaller as well, but it approaches zero, rather than negative infinity.  


This behavior is characteristic of exponential functions with a base larger than 1.  If the base was between 0 and 1, then the behavior would be much different: y would tend towards zero as x gets larger, but tend toward infinity as x gets smaller. 

Let's contrast this with the graph of an exponential equation with a larger base, such as y = 3x

We have the same general behavior as in the previous graph, however things are more dramatic: as x becomes more positive, y increases, but at a faster rate than in the previous graph.  This is because a larger base number is being raised to a positive exponent.  Similarly, as x becomes more negative, y decreases in value (approaching zero), but at a faster rate than in the previous graph.  This is we can think of y being divided by a larger number each time x decreases in value. 

The y-intercept of y = abx

The y-intercept to any equation is the point on the graph at which the line or curve touches or crosses the y-axis. This always occurs when x = 0.  Let's return to the equation y = 2x  When x = 0, y evaluates to 1, because any base number raised to a power of zero is 1.  That must mean this holds true for the equation y = 3x  The y-intercept is at the point (0, 1) because when x = 0, y = 1 for any exponential equation in the form y = bx

What is the y-intercept of equations in the form y = abx?  We alreay know that bx evalutes to 1 when x = 0 for any base.  We can deduce that the y-intercept depends on the value of "a" in this case.  For the general exponential equation y = abx, the y-intercept has the coordinates (0, a)

Big Idea

  • For equations in the form y = bx, the y–intercept has coordinates (0, 1) because b0 = 1 for any accepted base. 
  • For equations in the form y = abx, the y–intercept has coordinates (0, a), for the same reasons above. 

Other Features of the Graph of y = abx

If we reverse the signs of a and x in the general exponential equation, we end up with different variations of the general exponential curve.  More specifically, these are reflections about either the x– or y–axes, or perhaps both, if the signs of both a and x are reversed. These patterns are illustrated in the graphs below: