11th Grade Q1

To determine the asymptote of an exponential function, you identify the horizontal line that the graph approaches but never touches. This asymptote reflects the long-term behavior of the function as $x$ becomes very large or very small.

An exponential function typically has the form:

where:

• $a$ is the constant multiplier,
• $b$ is the base of the exponential,
• $x$ is the variable (exponent),
• $c$ is the vertical shift.

The asymptote is determined by the constant $c$ in the equation.

Method

1. Identify the horizontal shift

In an exponential function, the term $c$ shifts the graph vertically. As $x\to \infty$ or $x\to -\infty$, the exponential term $a\cdot b^x$ approaches $0$. Thus, the function approaches the value of $c$.

Horizontal asymptote:

The horizontal asymptote is the line:

2. Behaviour of the graph

• If $a>0$ and $b>1$, the graph grows as $x\to \infty$ and approaches $y=c$ as $x\to -\infty$.
• If $a>0$ and $0<b<1$, the graph decays as $x\to \infty$ and approaches $y=c$ as $x\to \infty$.
• If $a<0$, the graph flips and approaches $y=c$ from below.

Example

Let’s determine the asymptote of $f(x)=3\cdot 2^x-4$.

1. The constant $c$ is $-4$.
2. As $x\to \infty$ or $x\to -\infty$, the term $3\cdot 2^x$ approaches $0$.
3. Therefore, the horizontal asymptote is:

You mark an asymptote on a plot by a dotted line. The plot of $f(x)=3\cdot 2^x-4$ looks like

Key Points to Remember

• The asymptote of an exponential function is given by the constant $c$ in the equation $f(x)=a\cdot b^x+c$.
• The function never crosses its asymptote but gets infinitely close to it as $x$ grows large or small.
• The horizontal asymptote reflects the long-term behaviour of the graph.