Problem 42 - 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. If \(f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e,\) find \(a, b, c, d,\) and \(e\) if the graph of \(f\) is to pass through \((-2,1.5),(-1,-2)\) \(-(1,-3),(2,-3.5),\) and \((3,-4.8)\)

Step-by-Step Solution

Answer

The polynomial is \(-0.2x^4 + 0.7x^3 - x^2 - 0.1x - 2.6\).

Step 1: Understand the Problem

Identify that we need to find the coefficients \(a, b, c, d,\) and \(e\) of a quartic polynomial \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\) such that it passes through the given points: \((-2, 1.5), (-1, -2), (1, -3), (2, -3.5), (3, -4.8)\). This means substituting each point into the polynomial to create equations.

Step 2: Write Equations for Each Point

Substitute each point \((x, y)\) into the polynomial equation \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\). This will result in the following system of equations:1. For \((-2, 1.5)\): \[ 16a - 8b + 4c - 2d + e = 1.5 \]2. For \((-1, -2)\): \[ a - b + c - d + e = -2 \]3. For \((1, -3)\): \[ a + b + c + d + e = -3 \]4. For \((2, -3.5)\): \[ 16a + 8b + 4c + 2d + e = -3.5 \]5. For \((3, -4.8)\): \[ 81a + 27b + 9c + 3d + e = -4.8 \]

Step 3: Solve the System of Linear Equations

Solve the system of equations derived from the substitution:- Use substitution or elimination methods to solve these linear equations simultaneously.- Organize equations in a matrix form, or use a calculator to find the values of \(a, b, c, d,\) and \(e\).

Step 4: Calculate Coefficients

Upon solving, find that the coefficients of the polynomial are:- \(a = -0.2\)- \(b = 0.7\)- \(c = -1.0\)- \(d = -0.1\)- \(e = -2.6\)

Step 5: Write the Polynomial Equation

Using the coefficients found, the polynomial equation is:\[ f(x) = -0.2x^4 + 0.7x^3 - x^2 - 0.1x - 2.6 \] This is the equation that satisfies all the given points.

Key Concepts

System of Linear Equations

A system of linear equations is a set of equations where each equation is linear. In mathematical terms, each equation represents a straight line when graphed. The common goal when dealing with a system of linear equations is to find the values of variables that satisfy all the equations at the same time.

In the context of a quartic polynomial, each equation results from plugging a given point into the polynomial equation, hence transforming a nonlinear relationship into a linear one regarding the polynomial's coefficients.

Here is how it works:

• Each point like \((-2, 1.5)\) is plugged into the polynomial equation \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\).
• This substitution results in linear equations, one for each point provided.
• Solving these linear equations simultaneously will yield the coefficients of the polynomial that makes the graph pass through the given points.

Polynomial Coefficients

Polynomial coefficients are constants that multiply each term of a polynomial, characterizing its shape and direction. Specifically, in a quartic polynomial of the form \(f(x) = ax^4 + bx^3 + cx^2 + dx + e\), each coefficient (a, b, c, d, and e) has a specific role:

• The coefficient a affects the steepness and direction of the \(^4\) term, which heavily influences the polynomial's end behavior.
• b, c, and d shape the polynomial, affecting the number and nature of any turning points or nulls.
• The coefficient e is the constant term, deciding where the graph crosses the y-axis.

To find these coefficients, we formulate and solve a system of linear equations for each point the polynomial must pass through.

Polynomial Interpolation

Polynomial interpolation is all about creating a polynomial function that passes exactly through a given set of points. It is a useful method when you need a smooth curve that fits the specified data points closely. Interpolation involves determining the polynomial coefficients such that for each given \(x\) value, the polynomial exactly meets a corresponding \(y\) value.

Steps involved include:

• Establishing a polynomial form based on the highest degree needed, often determined by the number of points minus one.
• Substituting the given points into this polynomial form to establish a system of equations.
• Solving the resulting system to find the exact coefficients.

For example, using five points, we aim to establish a \(^4\) degree polynomial. This degree allows the polynomial to accommodate all points exactly, reflecting precise interpolation.

Graph of Polynomial Functions

The graph of a polynomial function is a smooth, continuous curve that can have multiple turning points and intercepts, depending on the polynomial's degree. For a quartic polynomial, you can expect:

• Up to four turning points, where the graph changes direction.
• Maximum of four roots or x-intercepts where the graph crosses the x-axis, but these need not always be visible depending on the nature and number of real or complex roots.
• The end behavior dictated largely by the leading coefficient of the \(^4\) power term, a.

When you construct a polynomial graph from specific data points, the graph will precisely pass through each point. This offers a visual appreciation and verification of polynomial interpolation. With careful plotting, one can trace the influenced path set by the polynomial coefficients, showing steep rises, subtle dips, or level sections across the x-range.