Question
What is the end behavior of the graph of g(x) = (x + 1)(x - 2)(x + 5)?
Answers
GPT 3.5
To determine the end behavior of the graph of g(x) = (x + 1)(x - 2)(x + 5), we need to consider the leading term in the expansion of the function.
First, let's expand the function:
g(x) = (x + 1)(x - 2)(x + 5)
= (x^2 - x - 2)(x + 5)
= (x^2 - x - 2)(x) + (x^2 - x - 2)(5)
= x^3 - x^2 - 2x + 5x^2 - 5x - 10
= x^3 + 4x^2 - 7x - 10
The leading term in the expansion is x^3.
For x→-∞ (as x approaches negative infinity), the leading term dominates the function. Since the leading term has an odd degree, the end behavior will be negative infinity.
For x→+∞ (as x approaches positive infinity), the leading term still dominates the function. Since the leading term has a positive coefficient and an odd degree, the end behavior will be positive infinity.
Therefore, the end behavior of the graph of g(x) = (x + 1)(x - 2)(x + 5) is as follows:
- As x approaches negative infinity, g(x) approaches negative infinity.
- As x approaches positive infinity, g(x) approaches positive infinity.
First, let's expand the function:
g(x) = (x + 1)(x - 2)(x + 5)
= (x^2 - x - 2)(x + 5)
= (x^2 - x - 2)(x) + (x^2 - x - 2)(5)
= x^3 - x^2 - 2x + 5x^2 - 5x - 10
= x^3 + 4x^2 - 7x - 10
The leading term in the expansion is x^3.
For x→-∞ (as x approaches negative infinity), the leading term dominates the function. Since the leading term has an odd degree, the end behavior will be negative infinity.
For x→+∞ (as x approaches positive infinity), the leading term still dominates the function. Since the leading term has a positive coefficient and an odd degree, the end behavior will be positive infinity.
Therefore, the end behavior of the graph of g(x) = (x + 1)(x - 2)(x + 5) is as follows:
- As x approaches negative infinity, g(x) approaches negative infinity.
- As x approaches positive infinity, g(x) approaches positive infinity.