Question

How are the graphs `f(x)=x^3` and `g(x)=0.75(x+1)^3` related?

Answer 1

The graph of `g(x)` can be described as a transformation of the graph of `f(x)`.

Explanation:

First, take a look at the graph of `f(x)`.
graph{x^3 [-10, 10, -5, 5]}

Now look at the graph of `g(x)`.
graph{0.75(x+1)^3 [-10, 10, -5, 5]}

The graph of `f(x)` is vertically shrunk by a factor of `0.75` and translated left 1 unit to obtain the graph of `g(x)`.

Answer 2

See below

Explanation:

`f(x)` is what we call the parent function. Let's see what `x^3` looks like:

graph{x^3 [-10, 10, -5, 5]}

Now let's see what `(x+1)^3` looks like:

graph{(x+1)^3 [-10, 10, -5, 5]}

You should see that it's just a shifted version of the parent function. In this case, it's a horizontal shift `1` unit to the left.

Now, let's see what the constant in the function does. Since it's hard to see what's going on if we use `0.75`, let's graph another function that will tell us what a constant THAT is LESS THAN `1` does. Let's look at `0.1(x+1)^3`

graph{0.1(x+1)^3 [-10, 10, -5, 5]}

You should see that the constant is the `y`-intercept - the value when it crosses the `x` axis.