MATH SOLVE

4 months ago

Q:
# What are the key features of logarithmic and exponential graphs and how do transformations affect them?

Accepted Solution

A:

What are the key features of logarithmic and exponential graphs?

The exponential function, is formally known as the real function ex, where e is Euler's number, approximately 2.71828; this function has as a domain of definition the set of real numbers, and has the peculiarity that its derivative is the same function.

The logarithm, on the other hand, is the inverse function of the exponential.

how do transformations affect them?

Let's see this question with two examples

logarithm

f (x) = lnx original function

f (x) = - lnx reflection in x axis

f (x) = ln (x-2) displacement two units to the right.

f (x) = 2 + lnx displacement two units up

f (x) = ln (-x) refect in axis y

f (x) = ln (2-x) displacement two units to the right, reflection y

f (x) = - ln (-x) reflection in x, and in y axis

Exponential

f (x) = 5 ^ x original function.

f (x) = - 5 ^ x reflection on x axis

f (x) = 5 ^ -x reflection in axis y

f (x) = 5 ^ (x + 3) horizontal displacement 3 units to the left

f (x) = 5 ^ x +3 vertical displacement 3 units towards the top

f (x) = 5 ^ (x + 1) -4 displacement 1 unit to the left and 4 units to the bottom.

The exponential function, is formally known as the real function ex, where e is Euler's number, approximately 2.71828; this function has as a domain of definition the set of real numbers, and has the peculiarity that its derivative is the same function.

The logarithm, on the other hand, is the inverse function of the exponential.

how do transformations affect them?

Let's see this question with two examples

logarithm

f (x) = lnx original function

f (x) = - lnx reflection in x axis

f (x) = ln (x-2) displacement two units to the right.

f (x) = 2 + lnx displacement two units up

f (x) = ln (-x) refect in axis y

f (x) = ln (2-x) displacement two units to the right, reflection y

f (x) = - ln (-x) reflection in x, and in y axis

Exponential

f (x) = 5 ^ x original function.

f (x) = - 5 ^ x reflection on x axis

f (x) = 5 ^ -x reflection in axis y

f (x) = 5 ^ (x + 3) horizontal displacement 3 units to the left

f (x) = 5 ^ x +3 vertical displacement 3 units towards the top

f (x) = 5 ^ (x + 1) -4 displacement 1 unit to the left and 4 units to the bottom.