Describe the end behavior of the graph of the polynomial function by completing these statements: f(x)→? as x→-∞ and f(x)→? as x→+∞.
1. f(x)=-(x^2)+1
2. f(x)==10x^3
(1) f(x)=-(x²)+1
is a parabola concave downwards, with a y-intercept of 1.
Thus the behaviour toward both infinities are -∞.
2. f(x)=-10x³ (note correction of typo)
is a cubic polynomial that starts from +∞ when x->-∞, and finishes off at -∞ as x->∞.
1. For the polynomial function f(x) = -(x^2) + 1, as x approaches negative infinity (x → -∞), the function approaches positive infinity (f(x) → +∞). As x approaches positive infinity (x → +∞), the function approaches negative infinity (f(x) → -∞).
2. For the polynomial function f(x) = 10x^3, as x approaches negative infinity (x → -∞), the function also approaches negative infinity (f(x) → -∞). As x approaches positive infinity (x → +∞), the function approaches positive infinity (f(x) → +∞).
To determine the end behavior of a polynomial function, we need to examine the leading term of the polynomial.
1. For the function f(x) = -(x^2) + 1, the leading term is -x^2. The degree of this function is 2, and since the leading coefficient is negative, the end behavior can be described as follows:
- As x approaches negative infinity (x → -∞), the function f(x) decreases without bound, meaning that the y-values of the function decrease indefinitely. So, we can write f(x) → -∞ as x → -∞.
- As x approaches positive infinity (x → +∞), the function f(x) also decreases without bound, but in the opposite direction. The y-values of f(x) decrease and approach negative infinity. Therefore, we can write f(x) → -∞ as x → +∞.
2. For the function f(x) = 10x^3, the leading term is 10x^3. The degree of this function is 3, and the leading coefficient is positive. Consequently, the end behavior can be described as follows:
- As x approaches negative infinity (x → -∞), the function f(x) increases without bound, meaning that the y-values of the function increase indefinitely. Thus, we can write f(x) → +∞ as x → -∞.
- As x approaches positive infinity (x → +∞), the function f(x) also increases without bound. The y-values of f(x) increase and approach positive infinity. Therefore, we can write f(x) → +∞ as x → +∞.