Graph of an Exponential Equation

1. The Graph of y = a· b^x

To learn more about the graph of the equation , let's look at a specific case.

EXAMPLE

Below is a sketch of the graph . Notice that the base b to the exponential expression is 2, and there is an implied scalar multiplier of 1 in front of this expression, which would be the a in the equation:



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.

EXAMPLE

Consider the graph .



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.

hint

Remember, the base can never be negative (less than 0.)

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2. The y-intercept of y = b^x vs. y = a· b^x

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 .

EXAMPLE

Let's return to the equation . When x equals 0, y evaluates to 1, because any base number raised to a power of zero is 1. So we can say when x equals 0, . That must mean this also holds true for the equation from the second example above.

For the equation , the y-intercept is at the point (0, 1) because when x equals 0, y equals 1.

But what about the y-intercept of equations in the form ? We already know that evaluates to 1 when x equals 0 for any base. We can deduce that the y-intercept depends on the value of in this case.

For the general exponential equation , the y-intercept has the coordinates

big idea

• For equations in the form , the y-intercept has coordinates (0, 1).
• For equations in the form , the y-intercept has coordinates

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3. Features of the Graph of y = a · b^x

The general exponential equation has a positive and a positive x. If we reverse the signs of 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 and x are reversed. These patterns are illustrated in the graphs below:

Let's look at the characteristics of each case.

3a. Positive a, Positive Exponent

Having a positive and a positive exponent is the general exponential equation:

Suppose we have the equations . In this graph, as x is tending toward positive infinity, y goes to positive infinity, and as x goes to negative infinity, y approaches 0.

3b. Positive a, Negative Exponent

Let's take a further look into the next comparison of having a positive or negative exponent, but still having a positive

Suppose now we have the equation . A similar equation to , but this equation has a negative exponent. We can see that since negative x and positive x are opposite, their graphs have opposite effects. When our exponent is negative, as x approaches positive infinity, now y is approaching 0, and as x is approaching negative infinity, then our y is approaching positive infinity.

3c. Negative a, Positive Exponent

The next characteristic we'll look at is comparing the equations and graphs of exponential equations when the value of the number in front of the base, is positive or negative.

Suppose we still have our original exponential equation . Let's compare that to the equation and graph of . In the graphs, we can see that it looks like the graph is reflected over the x-axis. When the value is negative, as the x-values approach positive infinity, the y-values approach negative infinity. It's decreasing instead of increasing. Also, notice that as the x-values approach negative infinity, the y-values are still approaching 0 as they were with a graph of our equation with a positive value.

3d. Negative a, Negative Exponent

Finally let's look at the characteristics of exponential equations that have both a negative value and a negative exponent.

Suppose now we have the equations and . This second equation has both a negative value and a negative exponent. When the equation has a negative and negative exponent, as the x-values approach positive infinity, we see that the y-values approach 0, and as the x-values approach negative infinity, we see that the y-values are also approaching negative infinity.

big idea