Math Practice
Every question about an exponential graph comes back to a handful of features: the horizontal asymptote, the y-intercept, the range, and whether the curve rises or falls. These twelve problems make you read each one straight from the equation, including after shifts and reflections.
What is the horizontal asymptote of
The parent
What is the y-coordinate of the y-intercept of
Set
What is the domain of
Any real number can be an exponent, so the domain is all real numbers,
Is
The base 0.4 is between 0 and 1, so each step to the right multiplies the output by 0.4 — the graph falls. That is exponential decay.
What is the range of
The parent range is
Find the y-coordinate of the y-intercept of
At
What is the range of
The graph of
In
Find the x-intercept of
Set
Find the x-intercept of
Set
Find the y-coordinate of the y-intercept of
At
What is the range of
Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work
Use the graphing exponential functions lesson for the full method. Every problem below can be answered from the equation alone, without plotting a table of points.
| Feature | |
|---|---|
| Domain | |
| Range | |
| y-intercept | |
| Horizontal asymptote | |
| Direction | rises if |
The asymptote is
For the y-intercept, substitute
Only two: a vertical shift k, which moves the asymptote to y = k, and a negative multiplier a, which flips the range to the other side of that asymptote. A horizontal shift slides the curve sideways and leaves the range alone.
Because any real number can be used as an exponent — positive, negative, zero or fractional. No input is ever excluded, so shifts and stretches cannot change the domain.
−2ˣ means −(2ˣ): the power is applied first, then the negative, giving a reflection of 2ˣ across the x-axis. (−2)ˣ has a negative base and is not an exponential function at all.
No. The parent graph y = bˣ never touches the x-axis. An x-intercept only appears after a shift moves the asymptote so that the curve has to cross y = 0, as in y = 2ˣ − 8.
Look at which way the curve goes as x increases. Rising to the right is growth (base above 1); falling toward the asymptote is decay (base between 0 and 1). A reflection across the x-axis reverses the picture, so check the sign of a as well.