We can transform graphs of exponential functions and logarithmic functions in the same way we transform any other functions. Let's review some of the more important transformations.
Notation
Transformation
Examples
f(x)→ f(x-h)
Horizontal Translation Shifts the graph left if h<0, or right if h>0
g(x) = 4^(x -3), shifts the graph of f(x) = 4^x to the right by 3 units.
g(x) = 4^(x +2), shifts the graph of f(x) = 4^x to the left by 2 units.
f(x)→ f(x)+k
Vertical Translation Shifts the graph up if k>0, or down if k<0
g(x) = log_2(x) + 5, shifts the graph of f(x) = log_2(x) up by 5 units.
g(x) = log_2(x) - 4, shifts the graph of f(x) = log_2(x) down by 4 units.
f(x) → f(- x) [5.5em]
f(x) → - f( x)
Reflection Flips the graph over x-axis if using - x as argument or over y-axis if using - f(x)
g(x) = 4^(- x), reflect the graph of f(x) = 4^x in the y-axis
g(x) = -4^x, reflect the graph of f(x) = 4^x in the x-axis
f(x)→ af(x)
Vertical Shrink or Stretch Graph stretches away from the x-axis if |a|>1 or shrinks toward the x-axis if 1<|a|<0
g(x) = 4log_2(x), stretch the graph of f(x) = log_2(x) by a factor of 4.
g(x) = 12log_2(x), shrinks the graph of f(x) = log_2(x) by a factor of 12.
Note, however, that a logarithmic function is only defined when its argument takes values within the interval (0,+ ∞). This will remain being the case after applying the transformations.
