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?

1 Answer
Jun 12, 2018

g(x) = -e^x-5
h(x) -e^x+5
C) g(x) and h(x) are translated version of the same function.

Explanation:

In general, given a function f(x), you:

  • 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:

f(x) \to f(x)+k \to -(f(x)+k) = -f(x)-k

Reflect and then shift:

f(x) \to -f(x)\to -f(x)+k

So, in your case, A) leads to

g(x) = -e^x-5

while B) leads to

h(x) = -e^x+5

which means that A) is the function -e^x translated 5 units down, while B) is the function -e^x translated 5 units up.

This means that the two functions are the same graph, translated at 10 units of distance.