Problem 40 - Precalculus: Functions and Graphs

Question

Find an equation of the cuble polynomial \(f(x)=a x^{3}+b x^{2}+c x+d\) that passes through the given points. $$P(0,4), \quad Q(1,2), \quad R(-1,10), \quad S(2,-2)$$

Step-by-Step Solution

Answer

The cubic polynomial is \(f(x) = -x^3 + 2x^2 - 3x + 4\).

Step 1: Write the Polynomial Equation

The general form of the cubic polynomial equation is \( f(x) = ax^3 + bx^2 + cx + d \). Since we have four points, we can create four equations by substituting the \(x\) and \(f(x)\) (or \(y\) values) into the polynomial.

Step 2: Substitute the Points into the Equation

Substitute each point into the equation: 1. For \(P(0, 4)\), we have: \(4 = a(0)^3 + b(0)^2 + c(0) + d\) which simplifies to \(d = 4\).2. For \(Q(1, 2)\), we have: \(2 = a(1)^3 + b(1)^2 + c(1) + d\) which simplifies to \(a + b + c + d = 2\).3. For \(R(-1, 10)\), we have: \(10 = a(-1)^3 + b(-1)^2 + c(-1) + d\) which simplifies to \(-a + b - c + d = 10\).4. For \(S(2, -2)\), we have: \(-2 = a(2)^3 + b(2)^2 + c(2) + d\) which simplifies to \(8a + 4b + 2c + d = -2\).

Step 3: Solve the System of Equations

Using the equations from Step 2: 1. \(d = 4\).2. \(a + b + c + d = 2\) becomes \(a + b + c + 4 = 2\), or \(a + b + c = -2\).3. \(-a + b - c + d = 10\) becomes \(-a + b - c + 4 = 10\), or \(-a + b - c = 6\).4. \(8a + 4b + 2c + d = -2\) becomes \(8a + 4b + 2c + 4 = -2\), or \(8a + 4b + 2c = -6\).We now solve this system for \(a\), \(b\), and \(c\).

Step 4: Subtract Equations to Find Relationships

Subtract equation 2 from 3: \((-a + b - c = 6) - (a + b + c = -2)\) gives\(-2a - 2c = 8\) or \(a + c = -4\).Subtract equation 2 from 4: \((8a + 4b + 2c = -6) - (a + b + c = -2)\) gives\(7a + 3b + c = -4\).

Step 5: Solve for One Variable

From \(a + c = -4\), we can express \(c\) in terms of \(a\): \(c = -4 - a\).Substitute \(c = -4 - a\) into \(a + b + c = -2\) to get \(a + b - 4 - a = -2\)Thus, \(b - 4 = -2\) gives\(b = 2\).

Step 6: Solve for Remaining Variables

Use \(c = -4 - a\) and \(b = 2\) in one of the previous equations:From \(-a + b - c = 6\), substitute the known values:\(-a + 2 - (-4 - a) = 6\) which simplifies to\(-a + 2 + 4 + a = 6\), leading to\(6 = 6\), so this confirms our values are consistent.Use another equation to find \(a\):\((7a + 3b + c = -4)\), substitute \(b = 2\) and \(c = -4 - a\):\(7a + 3(2) + (-4 - a) = -4\) simplifies to\(6a + 6 - 4 = -4\),\(6a + 2 = -4\),\(6a = -6\),\(a = -1\).Then, using \(c = -4 - a\): \(c = -4 + 1 = -3\).

Step 7: Write the Final Equation

Using the values \(a = -1\), \(b = 2\), \(c = -3\), and \(d = 4\), our cubic polynomial is:\(f(x) = -x^3 + 2x^2 - 3x + 4\).

Key Concepts

Systems of Equations

A system of equations is a collection of two or more equations with the same set of variables. In this kind of problem, we're dealing with four separate equations that come from substituting the coordinates of the points into the general cubic polynomial equation. The system of equations allows us to find the values of the unknown coefficients.

• Each point on the plane provides one equation when substituted into the polynomial.
• In our problem, substituting each point produces four equations because four points are given.
• To solve this system, you should find values for the variables that satisfy all equations simultaneously.

Using methods such as substitution or elimination, we gradually reduce and solve these equations. This process helps determine the specific coefficients for the polynomial equation, ensuring it passes through all given points. It is very similar to solving a puzzle with pieces fitting together neatly once every variable is determined.

Polynomial Functions

Polynomial functions are algebraic expressions that involve sums of powers of a variable. In our task, we are working with cubic polynomials, which are polynomials of degree three. This means that the highest power of the variable, in this case, is three. Such equations take the general form:
\[ f(x) = ax^3 + bx^2 + cx + d \]

• Each term in the polynomial represents a different influence on the graph's shape.
• The coefficients \(a\), \(b\), \(c\), and \(d\) determine how the polynomial behaves and its exact positioning.
• The degree of the polynomial affects its general shape, predicting how many direction changes (turns) the graph can have.

The goal of our problem is to find the specific cubic polynomial that exactly passes through the given points. Understanding polynomial functions helps you see how changes in coefficients affect the graph and solve real-world problems modeled by these expressions.

Algebraic Equations

Algebraic equations form the foundation of algebra, consisting of numbers and variables that are solved to find an unknown. In our exercise, algebraic methods help to solve for the variables involved in our system of equations.

• By substituting given values and simplifying, we convert broad expressions into specific equations.
• Algebraic manipulation, such as adding, subtracting, or rearranging terms, assists in simplifying complex systems.
• Ultimately, solving these equations involves pinpointing each coefficient, allowing our polynomial to fit the given data points precisely.

Applying algebraic equations in this task demonstrates their practical use, showcasing how variables and constants interact to describe real-world relationships through vivid graphical representation.