Find an expression for a cubic function f if f(2) = 80 and f(−6) = f(0) = f(3) = 0. A cubic function generally has the form f(x) = ax3 + bx2 + cx + d. If we know that for some x-value x = p we have f(p) = 0, then it must be true that x − p is a factor of f(x). Since we are told that f(3) = 0, we know that x-3 is a factor. Similarly, since f(−6) = 0, then f(x) has the factor x+6 , and since f(0) = 0, then f(x) has the factor x-0 Since f(x) has the factors x, x − 3, and x + 6, then we must have f(x) = a(x)(x − 3)(x + 6) for some as yet unknown multiplier a. However, since it was given that f(2) = 80, we substitute 2 for x and 80 for f(x) and solve: f(2) = 80 = a(2)(2 − 3)(2 + 6) to obtain a = -5 1. Using all the information we found, write the full expression (expand, write full expression): f(x) =??????
Find an expression for a cubic function f if f(2) = 80 and f(−6) = f(0) = f(3) = 0. A cubic function generally has the form f(x) = ax3 + bx2 + cx + d. If we know that for some x-value x = p we have f(p) = 0, then it must be true that x − p is a factor of f(x). Since we are told that f(3) = 0, we know that x-3 is a factor. Similarly, since f(−6) = 0, then f(x) has the factor x+6 , and since f(0) = 0, then f(x) has the factor x-0 Since f(x) has the factors x, x − 3, and x + 6, then we must have f(x) = a(x)(x − 3)(x + 6) for some as yet unknown multiplier a. However, since it was given that f(2) = 80, we substitute 2 for x and 80 for f(x) and solve: f(2) = 80 = a(2)(2 − 3)(2 + 6) to obtain a = -5 1. Using all the information we found, write the full expression (expand, write full expression): f(x) =??????
Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find an expression for a cubic function f if f(2) = 80 and f(−6) = f(0) = f(3) = 0.
A cubic function generally has the form f(x) = ax3 + bx2 + cx + d. If we know that for some x-value x = p we have f(p) = 0, then it must be true that x − p is a factor of f(x). Since we are told that f(3) = 0, we know that x-3 is a factor.
Similarly, since f(−6) = 0, then f(x) has the factor x+6 , and since f(0) = 0, then f(x) has the factor x-0
Since f(x) has the factors x, x − 3, and x + 6, then we must have f(x) = a(x)(x − 3)(x + 6) for some as yet unknown multiplier a. However, since it was given that f(2) = 80, we substitute 2 for x and 80 for f(x) and solve:
f(2) = 80 = a(2)(2 − 3)(2 + 6)
to obtain a = -5
1. Using all the information we found, write the full expression (expand, write full expression):
f(x) =??????
Expert Solution
Step 1
given information:
a=-5, and the factors of f(x) are x, (x+6) and (x-3)
hence, f(x)=a(x)(x+6)(x-3).
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