A polynomial of degree 4 that increases the y-axis at 6 and whose y-values get more and more negative as x -> +infinity and x -> -infinity
-Represent a function that satisfies the given conditions by an equation. Leave in factored form.
To represent a polynomial function that satisfies the given conditions, we can start by considering the key features of the function.
1. The function has a degree of 4, indicating that its highest power term is x^4.
2. The function increases as y-values get more and more negative as x approaches positive infinity and negative infinity. This means that the leading coefficient of the polynomial is negative.
Based on these conditions, we can write the equation of the polynomial function in factored form. Let's proceed step by step:
Step 1: Start with the basic form of a polynomial:
f(x) = a(x - r₁)(x - r₂)(x - r₃)(x - r₄)
Step 2: Identify the degree of the polynomial:
The degree is 4, so we know there will be four unique roots, r₁, r₂, r₃, and r₄.
Step 3: Determine the sign of the leading coefficient:
Since the function needs to increase as y-values become more negative, the leading coefficient should be negative. Let's assume a = -1 for now.
Step 4: Determine the y-axis intercept:
The function increases by 6 on the y-axis, meaning it crosses the y-axis at y = 6. Thus, we know that one of the factors is (x - 0) = x.
Step 5: Determine the behavior as x approaches ± infinity:
Since the y-values need to get more and more negative as x approaches ± infinity, the remaining factors should involve x - ∞ and x + ∞.
Putting it all together, we have:
f(x) = -1 * x * (x - ∞) * (x + ∞) * (x - r)
f(x) = -x * (x - ∞) * (x + ∞) * (x - r)
Note that we replaced the unknown roots with the variable "r" for simplicity.
This equation represents a polynomial function of degree 4 that satisfies the given conditions. Keep in mind that the actual value of "r" is yet to be determined.