Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

검증된 단계별 안내

Start by writing the general form of the cubic function: $f(x) = a x^{3} + b x^{2} + c x + d$, where $a$, $b$, $c$, and $d$ are constants to be determined.

Use the given values of the function to set up a system of equations by substituting each $x$ value into the function and equating it to the corresponding $f(x)$ value: - For $f(-1) = 0$: $a(-1)^{3} + b(-1)^{2} + c(-1) + d = 0$ - For $f(1) = 2$: $a(1)^{3} + b(1)^{2} + c(1) + d = 2$ - For $f(2) = 3$: $a(2)^{3} + b(2)^{2} + c(2) + d = 3$ - For $f(3) = 12$: $a(3)^{3} + b(3)^{2} + c(3) + d = 12$

Simplify each equation to express them in terms of $a$, $b$, $c$, and $d$: - $-a + b - c + d = 0$ - $a + b + c + d = 2$ - $8a + 4b + 2c + d = 3$ - $27a + 9b + 3c + d = 12$

Set up the system of four linear equations with four unknowns: \[\begin{cases}$ -a + b - c + d = 0 \\ a + b + c + d = 2 \\ 8a + 4b + 2c + d = 3 \\ 27a + 9b + 3c + d = 12 $\end{cases}\]

Solve this system using methods such as substitution, elimination, or matrix operations (like Gaussian elimination) to find the values of $a$, $b$, $c$, and $d$. Once found, substitute these values back into the general form to write the specific cubic function.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

25m

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Cubic Functions

A cubic function is a polynomial of degree three, generally written as f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0. Understanding its general form helps in setting up equations based on given function values.

Using Function Values to Form Equations

Given specific values of the function at certain points, you can substitute these x-values into the cubic function to create a system of equations. Each substitution yields an equation involving a, b, c, and d, which can be solved simultaneously.

Solving Systems of Linear Equations

To find the coefficients a, b, c, and d, you solve the system of linear equations formed by the function values. Techniques include substitution, elimination, or matrix methods, enabling determination of the unique cubic function fitting the given points.