Algebra—end behavior

Question

Find the end behavior of...f(x)=(x-1)(x-2)^2 and explain how

Answer 1

Step 1: In google, type in "Types of Functions". Find a picture that shows 8-10 different types of functions (x, x^2, x^3, 1/x, ln(x), etc....). If you memorize this, you will get A's on tests.

Step 2: Expand your equation f(x)=(x-1)(x-2)^2 by doing something like FOIL method. You should get f(x)=x^3 - 5x^2 + 8x - 4.

Step 3: When we are asked to figure out end behavior, we are thinking about what happens when the values of x go to infinite. Remember that chart in Step 1. Look at the graph for x^3. When we talk about end behavior we really just care about the largest polynomial (x^3) because that denotes the majority of the behavior of the function. For the graph of x^3, as x goes to infinite you see that y goes to infinite. Same with our function (x-1)(x-2)^2. It is a cubic function so it will go to infinite too. Same if we go to negative infinite on x. Our function goes to negative infinite.

Step 4: memorize the end behaviors of all those other graphs. A lot of algebra and algebra 2 have these sort of questions and it will make it a lot easier for you (less time doing homework = more time having fun). Trust me. It will save your life.

Answer 2

The tutor Umar is exactly right. Here is the same problem explained in different words.

It is given that f(x) = (x - 1) (x - 2)²

We are only interested in the largest power of x when it is multiplied out. That is x³. The other terms will have little influence on the graph when x goes to extreme values.

So for end behavior we want to know what SIGN f(x) will have when x is a very very large positive number and when x is a very very large negative number.

So first when x is a large positive number, x³ will be a positive number and thus graph above the x axis.

And when x is a large negative number, x³ will be a negative number and thus graph below the x axis.

So the end behavior tells you the graph is below the x axis and rising in the third quadrant, crossing the x axis maybe more than once, and rising in the first quadrant. End behavior does NOT tell you what is happening to the graph around the origin.

Answer 3

This type of polynomial is a cubic equation. If we expand everything in the parentheses, we get

f(x)=x^3 - 5x^2 + 8x - 4. Now we can examine the first term and take a look at how the function behaves as the input x goes to infinity or goes to negative infinity. x^3 does not change the sign from negative to positive. Now we find that as the value of x increases, the value of x^3 and therefore the value of f(x) increases as it goes towards the positive x direction and infinity. At the other end, as x decreases towards negative infinity, x^3 approaches negative infinity, and so does f(x).

Note that since f(x) is not equal to x^3 and has other terms, not all positive values of x have a positive value in f(x) and not all negative values of x have a negative value in f(x)