Question

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?

Answer 1

`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.