Characteristics of Polynomials

[Veronica_Oles]

Homework Statement

I have to sketch a graph of y=x(x-3)^2

Homework Equations

The Attempt at a Solution

I know that the zeros are 0 and 3. The part which confuses me is that end behaviours as well as turning points. I am unsure of which way the end behaviours should be pointing. Is the highest degree 2 or 3? And how to I know which quadrants it should be traveling to and from?

 

[Ray Vickson]

Homework Statement

I have to sketch a graph of y=x(x-3)^2

Homework Equations

The Attempt at a Solution

I know that the zeros are 0 and 3. The part which confuses me is that end behaviours as well as turning points. I am unsure of which way the end behaviours should be pointing. Is the highest degree 2 or 3? And how to I know which quadrants it should be traveling to and from?

Look at the intervals $x < 0$, $0< x < 3$ and $x > 3$ separately. Now, think a bit: what happens to $y$ when $x$ becomes large and $> 0$? What happens when $x$ becomes large in magnitude but $< 0$ (that is, large negative)?

 

[Mark44]

To expand on what Ray said concerning the x-intercepts, when x is "close to 0" the graph is "close to" y = x(0 - 3)² = 9x. In other words, near x = 0, the graph of your polynomial looks a lot like the graph of the line y = 9x.

When x is "close to" 3, the graph of your polynomial resembles y = 3(x - 3)², a parabola. I'm hopeful that you have a good idea about the shape of this parabola.

If you expand x(x - 3)², it should be obvious what the degree of this polynomial is.

 

[Veronica_Oles]

To expand on what Ray said concerning the x-intercepts, when x is "close to 0" the graph is "close to" y = x(0 - 3)² = 9x. In other words, near x = 0, the graph of your polynomial looks a lot like the graph of the line y = 9x.

When x is "close to" 3, the graph of your polynomial resembles y = 3(x - 3)², a parabola. I'm hopeful that you have a good idea about the shape of this parabola.

If you expand x(x - 3)², it should be obvious what the degree of this polynomial is.

I understand it now.