In this lesson, you will learn how to identify features of an exponential graph. Specifically, this lesson will cover:
1. The Graph of y = abx
To learn more about the graph of the equation , let's look at a specific case.
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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.
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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.
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Remember, the base can never be less than 0.
2. The y-intercept of y = bx vs. 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 .
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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
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- For equations in the form , the y-intercept has coordinates (0, 1).
- For equations in the form , the y-intercept has coordinates
3. Features of the Graph of y = abx
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.
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An equation in the form
is an exponential equation. The
graph of has certain characteristics. The
y-intercept of function is equal to the value of
in the equation. Other features of the graph include looking at cases when
is negative and when the exponent is negative. A negative exponent reflects the graph over the y-axis. while a negative
coefficient reflects the graph over the x-axis.