what does it mean to describe the end behavior of f in a graph?
What happens to the graph or where is the graph at the far right side or the far left side of the x-axis?
or
what happens to the function as x ---> ± ∞
what does ---> +or - infinity mean?
and what that for f(x)=(x^2+x+3)/(x-1)
and f(x)= 3*2^x
To describe the end behavior of a function, we examine the behavior of the graph as the input values (x) approach positive infinity (+∞) and negative infinity (-∞). This allows us to understand what happens to the function as x becomes extremely large or small.
To determine the end behavior of a function, follow these steps:
1. Examine the degree of the polynomial function:
- If the function is a polynomial of even degree, look at the leading term (the term with the highest power of x).
- If the leading term has a positive coefficient, the graph rises to positive infinity on both ends.
- If the leading term has a negative coefficient, the graph falls to negative infinity on both ends.
- If the function is a polynomial of odd degree, the end behavior differs depending on the sign of the leading term.
- If the leading term has a positive coefficient, the graph rises to positive infinity as x approaches +∞ and falls to negative infinity as x approaches -∞.
- If the leading term has a negative coefficient, the graph falls to negative infinity as x approaches +∞ and rises to positive infinity as x approaches -∞.
2. If the function has any other type (not a polynomial), consider the following possibilities:
- If the function grows without bound as x approaches +∞ or -∞, the end behavior is described as "increasing without bound" or "decreasing without bound," respectively.
- If the function approaches a horizontal asymptote as x approaches +∞ or -∞, describe the equation of the asymptote as well.
By analyzing the degree and coefficients of the function, you can determine how it behaves at infinity and describe the end behavior of the graph.