Determine the end behavior for f(x)=3x^3+x^2-1

Question

Advanced functions math

Answer 1

For end behavior of a graph, you only need to look at the leading term (term with highest exponent). Because end behavior is really talking about what happens when x gets REALLY big.

 

 

If you have a function g(x)=x²-7 when x is really big, say x= 1,000,000, g(1,000,000)= (1,000,000)²-7 With a number as large as (1,000,000)²=1,000,000,000,000 that -7 is not having much effect on the graph. So we can just look at the leading term for end behavior. In this example, the leading term is x² so this graph will have the same end behavior as x².

 

If you don't know the end behavior of your leading term, you can figure it out. 

 

***Left="as x approaches negative infinity" and right="as x approaches positive infinity"*** 

Terms with EVEN exponents

If the leading term has an even exponent, the ends of the graph will go in the same direction (either both up, or both down). To determine if they go up or down, you look at the leading coefficient.

leading coefficient:

positive: both up (y goes to positive infinity)

negative: both down (y goes to negative infinity)

This works because if you raise any number (positive or negative) to an even power, you get a positive number.

For example (-3)²=+9 and -2(-3)²=(-2)(9)= -18

Terms with ODD exponents
If the leading term has an even exponent, the ends of the graph will go in different directions (either down on left and up on right, or up on the left and down on the right). To determine which way they go, you look at the leading coefficient.
leading coefficient:
positive: down on left (y goes to negative infinity), up on the right (y goes to positive infinity) 
negative: up on the left (y goes to positive infinity), down on the right (y goes to negative infinity)

This works because if you raise a positive number to an odd power, you get a positive number. But if you raise a negative number to an odd power, you get a negative number.
For example  (-3)³=-27 (will go to negative infinity on the left)  and (3)³=27 (will go to positive infinity on the right)

 and (-2)(-3)³=54 (will go to positive infinity on the left) and (-2)(3)³=-54 (will go to negative infinity on the right)

 

So for yours f(x)=3x³+x²-1

 

the leading term is 3x³

The exponent is odd, so the will go in different directions. the Leading coefficient is positive so: down on left (negative infinity), up on the right (positive infinity) 

 

 

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