Question

Solve \(f(x)=g(x)\) by graphing and algebraic methods. \(f(x)=x^3-4 x^2+4 x ; g(x)=-2 x+4\)

Short answer

The solution to \(f(x) = g(x)\) can be found graphically by identifying the points where the two graphs intersect and algebraically by setting the functions equal to each other and then solving the resulting equation. The algebraic solution will likely require the use of numerical methods or approximation techniques.

Step-by-step solution

Graph the functions

Draw the graphs for the functions \(f(x) = x^3 - 4x^2 + 4x\) and \(g(x) = -2x + 4\) on the same coordinate plane. Look for points where the two functions intersect. The x-values of these points are the solutions to the equation \(f(x) = g(x)\).

Set the functions equal to each other

For the algebraic method, we set \(f(x)\) equal to \(g(x)\). We have \(x^3 - 4x^2 + 4x = -2x + 4\).

Simplify the equation

To simplify the equation, combine like terms. Our equation becomes \(x^3 - 4x^2 + 6x - 4 = 0\). This is a cubic equation.

Solve the cubic equation

Solving this equation algebraically can involve various methods such as factoring, using the rational roots theorem, or synthetic division. Given the nature of this problem, it's likely necessary to use numerical methods or approximation techniques for finding the roots.

Key concepts

Graphical Method

The graphical method involves plotting the functions on the same coordinate plane to find their intersection points.
This visual approach helps you see where two functions, such as polynomials and linear equations, meet. In this context, we plot the cubic function, \(f(x) = x^3 - 4x^2 + 4x\), and the linear function, \(g(x) = -2x + 4\).

To solve the equation \(f(x) = g(x)\) graphically, follow these steps:

• Draw the graph of \(f(x)\), which typically has a curve shape due to its cubic nature.
• Draw the graph of \(g(x)\), a straight line.
• Identify the points where the line intersects the curve, as these represent the x-values that satisfy \(f(x) = g(x)\).

These intersection points are the solutions to the equation.
The graphical method is excellent for gaining an intuitive understanding of where solutions exist and how many there are.
However, it might not provide the exact numerical solutions, especially for non-integer or complex roots.

Algebraic Method

The algebraic method is used to find exact solutions by manipulating equations with mathematical operations.
This approach can be especially useful when graphical methods aren't precise enough. In our problem, we start by equating \(f(x)\) and \(g(x)\):

• Set the expressions equal: \(x^3 - 4x^2 + 4x = -2x + 4\).
• Simplify by combining like terms: \(x^3 - 4x^2 + 6x - 4 = 0\).

After simplifying, you're left with a cubic equation that needs solving.
This process requires algebraic techniques such as factoring, using the rational root theorem, or synthetic division.
The algebraic method provides exact values for x but can involve complex calculations, especially when dealing with higher-degree polynomials.

Cubic Equations

A cubic equation is a polynomial equation of the form \(ax^3 + bx^2 + cx + d = 0\), where \(a eq 0\).
These equations can have up to three real roots, and their graphs are characterized by their distinctive curve.

In our case, the equation simplifies to \(x^3 - 4x^2 + 6x - 4 = 0\).
Cubic equations can be challenging to solve because they may not factor neatly or yield to simple roots.

Methods for solving them include:

• Factoring, which requires finding one or more roots that simplify the equation.
• The Rational Root Theorem, which helps identify possible rational roots using factors of the constant term \(d\) and the leading coefficient \(a\).
• Synthetic division, a simplified form of the long division used to test potential roots.
• Numerical methods, like guesstimation or computational tools, for cases where traditional methods prove difficult.

Cubic equations are more complex than quadratics, requiring a deeper understanding of polynomial behavior and relationships.

Intersection Points

Intersection points are critical in solving equations graphically, as they are the solutions to \(f(x) = g(x)\).
These are the x-values where two functions share the same y-value.

In our exercise, looking for these intersection points means finding the solutions to \(x^3 - 4x^2 + 4x = -2x + 4\).
Graphically:

• These are the coordinates where the line of \(g(x) = -2x + 4\) and the curve of \(f(x) = x^3 - 4x^2 + 4x\) meet.

Algebraically, we find these by solving the equality: \(x^3 - 4x^2 + 6x - 4 = 0\).
Solving for intersection points can reveal both simple and complex roots, depending on the nature of the functions involved.
The number of real intersection points corresponds to the number of real solutions of the equation.