Question

How do you sketch the general shape of `f(x)=x^3-4x^2+7` using end behavior?

Answer 1

Look below :)

Explanation:

Since the first power is odd the general shape of the graph is similar to that of `x^3`. So the end behavior is `x^3`. We can also find the y-intercept by replacing all the x-values with zeros:

`f(0)=(0)^3-4(0)^2+7`

`y=7`

All you really need to know to sketch the general shape is the know the end behavior and the y-intercept.

This is the graph of `x^3`
graph{x^3 [-10, 10, -5, 5]}

This is the graph of `y=x^3-4x^2+7`

As you can see the point `(0,7)` is the y-intercept.

graph{x^3-4x^2+7 [-11.29, 11.21, -3.11, 8.14]}