Transformations of Exponential and Logarithmic Functions
Vertical Shifts
A vertical shift moves the graph up or down without changing its shape.
- For \$y = f(x) + a\$, the graph shifts up by \$a\$ units.
- For \$y = f(x) - a\$, the graph shifts down by \$a\$ units.
The asymptotesof the graph also shift vertically. For exponential functions, the horizontal asymptote becomes \$y = a\$. For logarithmic functions, the vertical asymptote remains unchanged.
Horizontal Shifts
A horizontal shift moves the graph left or right.
- For \$y = f(x + b)\$, the graph shifts left by \$b\$ units.
- For \$y = f(x - b)\$, the graph shifts right by \$b\$ units.
For logarithmic functions, the vertical asymptoteshifts horizontally. If the function is \$y = \log_b(x + c)\$, the asymptote is at \$x = -c\$.
Self review1. Sketch the graph of \$f(x) = 3^x\$ and \$g(x) = 3^x - 4\$. Describe the transformation. 2. For \$h(x) = \log_3(x - 2)\$, identify the vertical asymptote and describe the transformation from \$f(x) = \log_3(x)\$.
Theory of KnowledgeHow do transformations of functions illustrate the idea of mathematical structure? Can these transformations be applied to other types of functions?
