## How does the graph of g(x) = 0.5(2)*-3 – 1 compare to the graph of the parent function f(x) = 2*?. Write a full

Question

How does the graph of
g(x) = 0.5(2)*-3 – 1
compare to the graph of
the parent function
f(x) = 2*?. Write a full
description.

in progress 0
3 months 2021-08-30T21:11:37+00:00 1 Answers 1 views 0

Here we have:

f(x) = 2^x

g(x) = 0.5*2^(x – 3) – 1

We want to compare g(x) and f(x).

The first thing we should do here, is to define the transformations used.

Vertical translation:

For a function f(x), a vertical translation of N units is written as:

g(x) = f(x) + N

if N > 0, the translation is upwards

if N < 0, the translation is downwards.

Horizontal translation:

For a function f(x), a horizontal translation fo N units is written as:

g(x) = f(x + N)

if N > 0, the translation is to the left

if N < 0, the translation is to the right.

Vertical dilation:

For a general function f(x), a vertical dilation of scale factor k is written as:

g(x) = k*f(x).

Ok, now let’s start with f(x), and try to use transformations to construct g(x).

If we start with a vertical dilation of scale factor k = 0.5, then:

g(x) = 0.5*f(x)

if now we apply a horizontal translation of 3 units to the right, we get:

g(x) = 0.5*f(x – 3)

if now we apply a vertical translation of 1 unit down, we get:

g(x) = 0.5*f(x – 3) – 1

Replacing by the actual function we get

g(x) = 0.5*2^(x – 3) – 1

So we got g(x).

Then, the graph of g(x) is the graph of f(x) dilated vertically by a scale factor of 0.5, then moved to the right 3 units, and then moved down one unit.