Introduction
An exponential function can change by a constant factor when its input increases by one. Its graph is shaped not only by the coefficient in front, but also by whether the base is greater than 1 or between 0 and 1. A small set of features—domain, range, intercepts, asymptote, and end behavior—lets us describe that graph precisely without needing to plot every point.
In this tutorial, we study functions of the form \(y=ab^x\), where \(a>0\) and \(b>0\). The case \(b=1\) needs to be treated separately: it gives a constant function, not a nonconstant exponential curve.
Definitions and the basic setup
For a positive base \(b\), the expression \(b^x\) is defined for every real number \(x\). Therefore, when \(a>0\) and \(b>0\), the function \(f:\mathbb{R}\to\mathbb{R}\) given by \(f(x)=ab^x\) has domain \(\mathbb{R}\). Because \(a\) and \(b^x\) are positive, every function value is positive.
Since \(b^0=1\), setting \(x=0\) gives \(f(0)=a\). Thus the graph always has y-intercept \((0,a)\). It has no x-intercept: for every real \(x\), \(ab^x>0\), so \(f(x)\) cannot equal zero.
For the range and end behavior, the key distinction is whether \(b>1\), \(0<b<1\), or \(b=1\). The base determines what happens as \(x\) moves toward either end of the real line.
Why the horizontal asymptote is \(y=0\) when \(b\ne1\)
When \(b>1\), positive powers become arbitrarily large as the exponent increases. Consequently, negative powers become arbitrarily small and positive, because \(b^{-x}=1/b^x\). This makes the function approach zero as \(x\) decreases without bound. At the other end, the function increases without bound.
When \(0<b<1\), write \(c=1/b\), so \(c>1\). The exponent law gives \(b^x=c^{-x}\). As \(x\) increases without bound, \(-x\) decreases without bound, and \(c^{-x}\) approaches zero. In the other direction, \(b^x\) increases without bound. These are the standard end behaviors of positive-base exponential functions; the reciprocal identity explains why the two cases reverse.
Here is why the range is all positive real numbers when \(b\ne1\). The function is continuous, and its end behavior takes it arbitrarily close to zero at one end and arbitrarily large at the other. Therefore, for any chosen \(r>0\), there are inputs where the function value is below \(r\) and inputs where it is above \(r\). By the Intermediate Value Theorem, the function takes the value \(r\) somewhere between those inputs. It takes no nonpositive values, so its range is exactly \((0,\infty)\).
The phrase “approaches zero” does not mean that the function reaches zero. Every value \(ab^x\) remains positive. The graph gets arbitrarily close to the line \(y=0\) at one end, without crossing or touching it.
Organizing the features
For \(b\ne1\), the complete feature list can be organized before making a sketch. The end behavior determines which side approaches the horizontal asymptote. The y-intercept anchors the graph at \(x=0\), while positivity rules out any x-intercept.
| Feature | Conclusion when \(a>0\), \(b>0\), \(b\ne1\) |
|---|---|
| Domain | \(\mathbb{R}\) |
| Range | \((0,\infty)\) |
| Y-intercept | \((0,a)\) |
| X-intercept | None |
| Horizontal asymptote | \(y=0\) |
The direction of the end behavior depends on the base. For \(b>1\), the graph approaches \(y=0\) on the left and rises without bound on the right. For \(0<b<1\), it rises without bound on the left and approaches \(y=0\) on the right.
Worked examples
Worked Example: Features of \(y=3\cdot2^x\)
Here \(a=3\) and \(b=2\). Both parameters are positive, and \(b>1\).
Domain: The base is positive, so every real exponent is allowed. The domain is \(\mathbb{R}\).
Range: Since \(b\ne1\) and \(a>0\), the range is \((0,\infty)\).
Intercepts: At \(x=0\), \(y=3\cdot2^0=3\cdot1=3\), so the y-intercept is \((0,3)\). Since \(3\cdot2^x>0\) for every \(x\), there is no x-intercept.
Asymptote and end behavior: Because \(b>1\), \(y=0\) is a horizontal asymptote as \(x\to-\infty\). The function approaches zero from above on the left and increases without bound as \(x\to\infty\).
Check with sample values: At \(x=-1\), \(y=3\cdot2^{-1}=3\cdot\frac12=\frac32\). At \(x=0\), \(y=3\). At \(x=1\), \(y=3\cdot2=6\). These values confirm the intercept and are consistent with the stated end behavior.
Worked Example: Features of \(y=8\left(\frac12\right)^x\)
Here \(a=8\) and \(b=\frac12\), so \(0<b<1\).
Domain: The base is positive, so the domain is \(\mathbb{R}\).
Range: Because \(b\ne1\), the range is \((0,\infty)\).
Intercepts: At \(x=0\), \(y=8\left(\frac12\right)^0=8\), so the y-intercept is \((0,8)\). The output is always positive, so there is no x-intercept.
Asymptote and end behavior: Here \(b=\frac12\) is between 0 and 1. The function approaches zero as \(x\to\infty\), so \(y=0\) is a horizontal asymptote on the right. As \(x\to-\infty\), the function increases without bound.
Check with sample values: At \(x=-1\), \(y=8\left(\frac12\right)^{-1}=8\cdot2=16\). At \(x=0\), \(y=8\). At \(x=1\), \(y=8\cdot\frac12=4\). The values move downward as \(x\) increases, and the right-side values head toward zero.
Worked Example: The special case \(y=5\cdot1^x\)
Here \(a=5\) and \(b=1\). Since \(1^x=1\) for every real \(x\), the function simplifies to \(y=5\).
Domain: Every real input is allowed, so the domain is \(\mathbb{R}\).
Range: The function has the single value 5, so the range is \(\{5\}\), not \((0,\infty)\).
Intercepts: At \(x=0\), \(y=5\), giving the y-intercept \((0,5)\). Since the output is never zero, there is no x-intercept.
Asymptote and end behavior: The graph is the horizontal line \(y=5\). Its value remains 5 as \(x\to\infty\) and as \(x\to-\infty\). Under the limit definition of horizontal asymptote, \(y=5\) is an asymptote at both ends, although the graph coincides with that line. The line \(y=0\) is not an asymptote.
Check: Substituting \(x=0\) gives \(5\cdot1^0=5\), and substituting \(x=2\) gives \(5\cdot1^2=5\). Both values agree with the constant function.
A common mistake: forgetting the base-one case
A frequent error is to state that every positive \(a\) and \(b\) gives range \((0,\infty)\). That statement fails when \(b=1\). For example, if \(a=5\) and \(b=1\), then \(5\cdot1^x=5\) for every \(x\); the function never takes the value 2, or any value other than 5.
A second useful check is not to confuse “positive outputs” with “all positive outputs.” Positivity alone proves that the range is contained in \((0,\infty)\). To conclude that the range equals \((0,\infty)\), we also need the nonconstant exponential’s end behavior and continuity, which ensure it takes every positive value.
A reliable feature-finding process
When given a function of this form, use the following order. First check whether \(b=1\); this prevents the constant case from being assigned the wrong range. Then identify the intercept and use the base to determine the end behavior.
Confirm that \(a>0\) and \(b>0\), then decide whether \(b=1\), \(b>1\), or \(0<b<1\).
The domain is \(\mathbb{R}\). The y-intercept is \((0,a)\), and there is no x-intercept because \(ab^x>0\).
If \(b=1\), the range is \(\{a\}\). Otherwise, the range is \((0,\infty)\); use the position of \(b\) relative to 1 to identify which end approaches zero.
For \(b\ne1\), it is \(y=0\) at the end where the function approaches zero. For \(b=1\), the graph is the line \(y=a\).
Why these features matter
These features give a dependable description of the graph before any detailed sketch. The intercept locates the graph at the vertical axis. The range and the absence of x-intercepts tell us the graph stays above the \(x\)-axis. The asymptote and end behavior explain how the graph behaves far from the origin.
They also provide checks on a proposed graph. A graph of \(ab^x\) with positive \(a\) and \(b\) cannot cross the \(x\)-axis. When \(b\ne1\), it must have positive outputs of every size and approach \(y=0\) at one end. When \(b=1\), it must be a horizontal line. If a sketch violates any of these facts, it cannot represent the stated function.
Check Your Understanding
For each question, use the parameter conditions carefully, including the distinction between \(b=1\) and \(b\ne1\).
- For \(y=4\cdot3^x\), state the domain, range, both intercept conclusions, horizontal asymptote, and end behavior.
- For \(y=6\left(\frac13\right)^x\), which end approaches the horizontal asymptote \(y=0\)?
- Why can \(y=ab^x\), with \(a>0\) and \(b>0\), never have an x-intercept?
- Find the range of \(y=7\cdot1^x\). Explain why it is not \((0,\infty)\).
- For \(y=2\cdot4^x\), compute the y-intercept by substituting \(x=0\), and state whether the function takes the value zero.
