How do you find a formula of a function given the function f(x)=e^x ?
Given the function f(x)=e^x
A) Find a formula for g(x) where the graph of g(x) is obtained from the graph of f(x) by shifting up 5 units and then reflecting about the x axis.
B) Find a formula for h(x) where the graph of h(x) is obtained from the graph of f(x) by reflecting about the x-axis and then shifting up 5 units.
C) Are the functions g(x) and h(x) the same? If not, how are their graphs related?
Given the function f(x)=e^x
A) Find a formula for g(x) where the graph of g(x) is obtained from the graph of f(x) by shifting up 5 units and then reflecting about the x axis.
B) Find a formula for h(x) where the graph of h(x) is obtained from the graph of f(x) by reflecting about the x-axis and then shifting up 5 units.
C) Are the functions g(x) and h(x) the same? If not, how are their graphs related?
1 Answer
C)
Explanation:
In general, given a function
- shift it vertically by adding constants:
#f(x) \to f(x)+k# - Reflect it about the
#x# axis by changing its sign:#f(x)\to -f(x)#
As you can see, the transformations are not commutative:
Shift and then reflect:
Reflect and then shift:
So, in your case, A) leads to
while B) leads to
which means that A) is the function
This means that the two functions are the same graph, translated at