Let's start by reviewing how can we transform the graph of an exponential function. Recall that we can do this the same way we do with other functions.
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) = 4^x + 5, shifts the graph of f(x) = 4^x up by 5 units.
g(x) = 4^x - 4, shifts the graph of f(x) = 4^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) = 4* 4^x, stretch the graph of f(x) = 4^x by a factor of 4.
g(x) = 12* 4^x, shrinks the graph of f(x) = 4^x by a factor of 12.
Now let's take a look to the given function.
ab^(x- h)+ k
Taking in to account what we already discussed, we know that the factor a shrinks or stretches the original exponential function vertically, the constant h shifts the function horizontally, and the constant k shifts the function vertically.
