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Overview

Transformations of exponential and logarithmic functions involve shifting, stretching, compressing, and reflecting their graphs. These transformations help us understand how changes to the function’s equation affect its graph.

Transformations Overview

The general transformations for exponential and logarithmic functions are:

• Vertical Shifts: $y=f(x)+k$: Shifts the graph up by $k$ if $k>0$, or down if $k<0$.
• Horizontal Shifts: $y=f(x-h)$: Shifts the graph right by $h$ if $h>0$, or left if $h<0$.
• Vertical Stretch/Compression: $y=a\cdot f(x)$: Stretches the graph vertically if $|a|>1$, compresses if $0<|a|<1$.
• Reflection: $y=-f(x)$: Reflects the graph across the x-axis. $y=f(-x)$: Reflects the graph across the y-axis.

Examples

Exponential Functions

The base function is $y=b^x$. Transformations affect it as follows:

• $y=2^x+3$: Shifts the graph up by 3.
• $y=2^{x-1}$: Shifts the graph right by 1.
• $y=-2^x$: Reflects the graph across the x-axis.
• $y=3\cdot 2^x$: Stretches the graph vertically by a factor of 3.

Logarithmic Functions

The base function is $y={\log }_b(x)$. Transformations affect it as follows:

• $y=\log (x)-2$: Shifts the graph down by 2.
• $y=\log (x+1)$: Shifts the graph left by 1.
• $y=-\log (x)$: Reflects the graph across the x-axis.
• $y=\frac{1}{2}\log (x)$: Compresses the graph vertically by a factor of 1/2.

Practice Questions

1. Question 1: Describe the transformation of $y=2^x-4$ compared to $y=2^x$.
   Solution

   The graph shifts down by 4 units.

2. Question 2: Write the equation for a logarithmic function that is shifted 3 units to the left and reflected across the x-axis.
   Solution

   $y=-\log (x+3)$

3. Question 3: The graph of $y=5^x$ is stretched vertically by a factor of 2 and shifted 1 unit up. Write the equation of the transformed function.
   Solution

   $y=2\cdot 5^x+1$

4. Question 4: The graph of $y=\ln (x)$ is reflected across the y-axis and compressed vertically by a factor of 0.5. Write the equation of the transformed function.
   Solution

   $y=0.5\ln (-x)$

5. Question 5: Graph the function $y=3^x-2$ and describe the transformation compared to $y=3^x$.
   Solution

   The graph is shifted down by 2 units.

Extra Practice

• Describe the transformation of $y=\log (x-4)+2$.
• Write the equation of an exponential function shifted 5 units up and reflected across the y-axis.
• Write the equation for a logarithmic function stretched vertically by 4 and shifted 2 units to the right.
• Find the transformations applied to $y={10}^x$ to obtain $y=-2\cdot {10}^{x+3}$.
• Sketch the graph of $y=\ln (x)-3$ and describe the transformation.