We want to show how the graph of the function y changes depending on c when a is positive and b≥ 1.
y= a b^(cx)
We will examine the effects of c in three cases.
When c>0
The effects of c, greater than zero, should be analyzed in three parts.
If 0< c<1 , the function grows slowly compared to the parent function of exponential functions. That is, the parent function is stretched horizontally.
If c= 1, we get the function y=ab^x.
y=ab^(1* x) ⇒ y=ab^x
If c>1, the function grows quickly compared to the parent function of exponential functions. That is, the parent function is compressed horizontally.
When c=0
The graph will be a constant function since the exponent will be zero.
y=ab^(0* x) ⇔ y=ab^0
By the Zero Exponent Rule, b^0=1, so the function becomes y=a.
y=a* 1 ⇔ y=a
When c<0
For the values of c less than 0, the function becomes a decreasing function and the graph is reflected across the y-axis.
Let's organize what we have said so far.
y=ab^(cx)
c - Orientation and Shape
If 0<|c|<1, the graph is stretched horizontally
If |c|>1, the graph is compressed horizontally
If c=0, the graph is equal to y=a
If |c|=1, the graph is equal to y=ab^x
If c<0, the graph is reflected across the y-axis
Consider the table above and examine the changes on the graph of the function for the different c-values.
