Answer:
a. f(0) = 7; x → ±∞, f(x) → +∞
b. f(0) = 9; x → -∞, f(x) → -∞; x → +∞, f(x) → +∞
Explanation:
You want the y-intercepts and end behavior of the functions ...
- f(x) = (x -7)(x -1)³
- f(x) = 4x⁵ +2x⁴ -8x +9
Y-intercept
The y-intercept is the constant in the function. It is the value when x=0. Substitute x=0 into the equation, and the value you get is the y-intercept.
End behavior
End behavior is determined by two things:
- the degree of the polynomial
- the sign of the leading coefficient
This tells you the only thing you need to look at is the leading term.
On a grand scale, even-degree polynomials have a graph that is U-shaped. Odd-degree polynomials have a graph that is /-shaped. These shapes are inverted when the sign of the leading coefficient is negative.
In every case, the value of a polynomial function is large (tends to infinity) when the value of the variable is large (tends to infinity). It is the signs of these infinities that are different in the different cases.
- even degree: the sign of the function value matches the sign of the leading coefficient.
- odd degree: the sign of the function value is the product of the sign of x and the sign of the leading coefficient.
F(x) = (x -7)(x -1)³
The y-intercept is ...
f(0) = (0 -7)(0 -1)³ = (-7)(-1)
f(0) = 7
The leading term is ...
f(x) ≈ (x)(x)³ = x⁴
This is even degree with a positive coefficient, so the end behavior is positive.
x → ±∞, f(x) → +∞
F(x) = 4x⁵ +2x⁴ -8x +9
The y-intercept is ...
f(0) = 4·0 +2·0 -8·0 +9
f(0) = 9
The leading term is ...
f(x) ≈ 4x⁵
This is odd degree with a positive coefficient, so the end behavior matches that of x.
x → -∞, f(x) → -∞; x → +∞, f(x) → +∞
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Additional comment
"End behavior" means "behavior for large (positive or negative) values of x." This is a vocabulary term that you need to remember.
If you think about what happens with large numbers, you realize that only the highest-degree term is important for very large values of x.
Consider the first two terms of the second function for x = 10^10. The value of the first term is 5·10^50. The value of the second term is 2·10^40, a value that is 10 orders of magnitude smaller. The sum of these two terms is 5.0000000002×10^50. We can effectively ignore the 4th and lower-degree terms for large x-values.
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