Graphs of Exponential and Logarithmic Functions
Exponential Functions
An exponential function has the form \(y = a^x\).
An ordinary exponential function always passes through:
since \(a^0 = 1\), \(a^1 = a\), and \(a^{-1} = \frac{1}{a}\).
The x‑axis is an asymptote. The graph never crosses the x‑axis.
When the base is greater than 1:
When the base is between 0 and 1:
To calculate values of \(y\), raise the base to the power \(x\).
For \(y = 2^x\):
Table of values
| x | y |
|---|---|
| -2 | 0.25 |
| -1 | 0.5 |
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
Logarithmic Functions
A logarithmic function always passes through:
since \(\log_a 1 = 0\) and \(\log_a a = 1\).
The y‑axis is an asymptote. The graph never crosses the y‑axis.
Sketch the graph of \(y=\log_{6}x\)
To calculate values of \(y\):
Table of values for \(y=\log_{6}x\)
| x | y |
|---|---|
| 0.2 | -1.113 |
| 0.5 | -0.386 |
| 1 | 0 |
| 2 | 0.386 |
| 3 | 0.613 |
| 4 | 0.773 |
| 6 | 1 |
| 10 | 1.285 |
| 20 | 1.773 |
| 36 | 2 |
Shifting Log Graphs Left and Right
Consider the graph \(y = \log x\):
The base is 10. Look for the points \((1,0)\) and \((10,1)\).
Now compare \(y = \log(x+2)\) and \(y = \log(x-2)\):
Note how they shift the opposite way!
Shifting Up and Down
The base is 10. Look for the points \((1,0)\) and \((10,1)\).
Shift upwards
\((1,0)\) moves to \((1,2)\) and \((10,1)\) moves to \((10,3)\).
Shift downwards
\((1,0)\) moves to \((1,-2)\) and \((10,1)\) moves to \((10,-1)\).
Bringing It All Together
The graph below has equation \(y = \log(x+a) + b\). Find the integers \(a\) and \(b\).
There is an asymptote at \(x = -3\). The graph has shifted left by 3 units.
So:
The base is 10. Normally the point \((10,1)\) exists.
Shift left 3 units → expect \((7,1)\).
But on the graph, when \(x = 7\), \(y = -1\). So the graph has moved down 2 units.
Thus:
