how do the degree and the sign of the leading coefficient correspond to the end behaviour of the polynomial function.
please help I don't know where to start..
http://www.purplemath.com/modules/polyends.htm
thank u
To understand how the degree and sign of the leading coefficient correspond to the end behavior of a polynomial function, start by considering the general form of a polynomial:
f(x) = a(nx^n) + a(n-1)x^(n-1) + ... + a1x + a0
- The degree of a polynomial is determined by the highest exponent, which is denoted as 'n' in the general form above.
- The leading coefficient is the coefficient of the term with the highest power, represented by 'a(n)'.
Now, let's discuss the different scenarios for the end behavior based on these two factors:
1. Even Degree Polynomial with Positive Leading Coefficient:
- When the degree of the polynomial is even (n is an even number) and the leading coefficient 'a(n)' is positive (+), the end behavior of the graph will be as follows:
- The graph approaches positive infinity as x approaches both positive infinity and negative infinity.
- This means the right end of the graph heads upwards and the left end also heads upwards.
2. Even Degree Polynomial with Negative Leading Coefficient:
- If the degree is even (n is an even number) but the leading coefficient 'a(n)' is negative (-), the end behavior is:
- The graph approaches negative infinity as x approaches positive infinity.
- The graph approaches positive infinity as x approaches negative infinity.
- In this case, both ends of the graph head in opposite directions: right end points downwards, and the left end points upwards.
3. Odd Degree Polynomial with Positive Leading Coefficient:
- When the degree of the polynomial is odd (n is an odd number) and the leading coefficient 'a(n)' is positive (+), the end behavior is:
- The graph approaches positive infinity as x approaches positive infinity.
- The graph approaches negative infinity as x approaches negative infinity.
- In this scenario, both ends of the graph head in opposite directions: right end points upwards, and the left end points downwards.
4. Odd Degree Polynomial with Negative Leading Coefficient:
- Finally, if the degree is odd (n is an odd number) and the leading coefficient 'a(n)' is negative (-), the end behavior of the graph is as follows:
- The graph approaches negative infinity as x approaches both positive infinity and negative infinity.
- Here, both ends of the graph head downwards.
Remember, the end behavior describes what happens to the graph as x approaches positive or negative infinity. By examining the degree and the sign of the leading coefficient, you can determine how the graph will behave at the ends or far off points on the x-axis.