Exponential Graph Basics

Statement

An exponential graph is any curve that can be written in the form:

\[ y = k a^x \quad (a>0,\, a\neq 1) \]

Here, \(a\) is called the base and \(k\) is a constant multiplier. All exponential graphs share the key property that the variable \(x\) appears in the exponent. These graphs have a horizontal asymptote at \(y=0\).

Why it works

• If \(a>1\), the function grows rapidly as \(x\) increases, showing exponential growth.
• If \(0
• Because \(a^x\) is always positive for real \(x\), the graph never touches or crosses the \(x\)-axis, which explains the horizontal asymptote at \(y=0\).

Recipe (Steps to Sketch/Use)

1. Identify the base \(a\): if greater than 1, growth; if between 0 and 1, decay.
2. Identify the multiplier \(k\): this scales the graph vertically and may reflect it if negative.
3. Plot key points such as \(x=0\) (where \(y=k\)), \(x=1\), and \(x=-1\).
4. Draw the asymptote at \(y=0\) as a dashed line.
5. Sketch the curve approaching the asymptote as \(x\to -\infty\) or \(x\to \infty\).

Spotting it

Exponential graphs appear in contexts such as population growth, radioactive decay, interest rates, and compound processes. Look for variables raised to a power of \(x\).

Common pairings

• Logarithmic graphs (inverses of exponential graphs).
• Compound interest and depreciation models.
• Half-life and doubling-time problems.

Mini examples

1. Given: \(y=2^x\). Answer: Growth, passes through (0,1), asymptote \(y=0\).
2. Given: \(y=(1/3)^x\). Answer: Decay, passes through (0,1), decreases to 0 as \(x\to\infty\).

Pitfalls

• Confusing base greater than 1 with decay — remember: \(a>1\) gives growth, \(0
• Forgetting that the graph never touches the asymptote.
• Assuming symmetry — exponential graphs are not symmetric.

Exam Strategy

• Always plot \(x=0\) to find \(y=k\).
• Check whether the graph is growth or decay by looking at \(a\).
• Use the asymptote \(y=0\) to guide the shape of your curve.
• If negative multiplier, reflect the graph in the \(x\)-axis.

Worked Micro Example

Sketch \(y=3(0.5)^x\).

Here, \(a=0.5\) (decay), \(k=3\). At \(x=0\), \(y=3\). At \(x=1\), \(y=1.5\). At \(x=2\), \(y=0.75\). The graph decreases towards asymptote \(y=0\).

Summary

Exponential graphs model many real-world processes involving repeated growth or decay. The base \(a\) controls whether the function grows or decays, and the constant \(k\) controls the initial value and vertical stretch. These graphs are always positive and approach, but never reach, the line \(y=0\).