An Introduction to Graphing Exponential Functions

Summary

The video introduces the graphs of exponential functions by plotting points and comparing how changes to the base affect the curve.

Graphing (f(x)=2^x)

The instructor builds a table of values and plots the corresponding points:

(x) (2^x) (-2) (\frac14) (-1) (\frac12) (0) (1) (1) (2) (2) (4) (3) (8)

These points show that the graph rises rapidly as (x) increases and approaches the (x)-axis as (x) decreases.

Key Properties of (2^x)

• The graph passes through ((0,1)), since any positive base raised to the zero power equals (1).
• Its values are always positive: (2^x) is never zero or negative.
• The (x)-axis, (y=0), is a horizontal asymptote. The graph approaches it on the left but never touches or crosses it.
• The function grows increasingly quickly to the right and declines toward zero to the left.

How the Base Affects the Graph

For bases greater than (1), the graph still passes through ((0,1)), but a larger base produces faster growth to the right.

For example, (3^x) is above (2^x) when (x>0). For negative (x), it is below (2^x) and approaches zero more quickly. For instance, (3^{-2}=\frac19), while (2^{-2}=\frac14). The larger base therefore makes the curve steeper on both sides of ((0,1)), while the (x)-axis remains an asymptote.

Bases Between 0 and 1

A function such as (\left(\frac12\right)^x) can be rewritten using exponent rules:

[ \left(\frac12\right)^x=2^{-x}. ]

Replacing (x) with (-x) reflects the graph of (2^x) across the (y)-axis. Thus, exponential functions with positive bases less than (1) decrease from left to right and approach zero on the right. The same reflection idea applies to (\left(\frac15\right)^x) and similar functions.

General Method

1. Choose several (x)-values, including zero, positive values, and negative values.
2. Evaluate the exponential expression carefully, using negative-exponent rules where needed.
3. Plot the resulting points.
4. Connect them with a smooth curve, noting the intercept, direction of growth or decay, and any horizontal asymptote.
5. To compare bases or graph a base between (0) and (1), use the relationships between the functions and their reflected shapes.

The video closes by previewing further examples involving negative numbers and transformations such as shifting graphs.

Speakers and Sources

• Presenter/instructor: An unnamed speaker explains the examples and graphing ideas.
• ThinkwellVids: The channel and source of the video.