Problem 23 - Algebra 2

Question

Determine whether a quadratic model exists for each set of values. If so, write the model. $$ f(-2)=16, f(0)=0, f(1)=4 $$

Step-by-Step Solution

Answer

Yes, a quadratic model exists for the set of values. The model is: \( f(x) = 4x^2 \).

Step 1: Identify the Task

First, understand that the task is to determine if a quadratic model, which is a second-degree polynomial of the form \( f(x) = ax^2 + bx + c \), can be fitted through the points (-2, 16), (0, 0), and (1, 4) given the function values.

Step 2: Set Up Equations Based on Given Points

Since \( f(x) = ax^2 + bx + c \) represents a quadratic model, substitute the given points into the equation to form a system of equations. For \( (-2, 16) \) the equation is \( 16 = a(-2)^2 + b(-2) + c \). For \( (0, 0) \) the equation is \(0 = a(0)^2 + b(0) + c \). And for \( (1, 4) \) the equation is \( 4 = a(1)^2 + b(1) + c \).

Step 3: Solve the System of Equations

Solve the system of equations obtained to find the values of \( a \) , \( b \) , and \( c \) that fit the quadratic model to the given set of values. The three equations are: \1) \( 16 = 4a - 2b + c \), \2) \( 0 = c \), \3) \( 4 = a + b + c \). From equation 2, we immediately get \( c = 0 \). Substitute \( c = 0 \) into equations 1 and 3 to solve for \( a \) and \( b \).

Step 4: Solve for a and b

With \( c = 0 \), the equations reduce to: \1) \( 16 = 4a - 2b \) and \2) \( 4 = a + b \). Solving the first equation for \( a \) in terms of \( b \) gives \( a = (16 + 2b)/4 \) or \( a = 4 + 0.5b \). Substituting \( a \) into the second equation yields \( 4 = (4 + 0.5b) + b \) which simplifies to \( 4 = 4 + 1.5b \) and further simplifies to \( b = 0 \). Plugging \( b = 0 \) back into the equation for \( a \) gives \( a = 4 \) .

Step 5: Write the Quadratic Model

Using the solved values for \( a \) , \( b \) , and \( c \), write the quadratic model which is \( f(x) = ax^2 + bx + c \). Given \( a = 4 \), \( b = 0 \), and \( c = 0 \), the quadratic model is \( f(x) = 4x^2 \).

Key Concepts

Systems of Equations

When dealing with systems of equations, like in the task to determine if a quadratic model can be created, we are talking about finding a set of values that satisfy multiple equations simultaneously. A system comprises two or more equations with a set number of variables. In this context, we'd be focusing on a system of three equations representing the quadratic model. The challenge is to find the particular values for the variables - in this case, the coefficients a, b, and c - that make all of the equations true at the same time.

Solving systems of equations can be approached using various methods, such as substitution, elimination, or graphical analysis. The exercise applied the substitution method effectively. This method involves expressing one variable in terms of the others and then substituting this into other equations, simplifying and solving.

• Substitute known values into the equations.
• Simplify and solve for one variable at a time.
• Back-substitute to find remaining variables.

The ease of solving a system of equations depends on the number of equations versus the number of variables. If they are equal, as in this case, you can typically find a unique solution.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0 where a, b, and c are coefficients, and a ≠ 0. The graph of a quadratic equation is a parabola which opens upward if a is positive and downward if a is negative.

In the given exercise, we were looking for a quadratic model that fits three points. This is a practical application of a quadratic equation because we are essentially trying to define the curve that passes through these points. The exercise walked through finding a, b, and c for the equation f(x) = ax^2 + bx + c using those points.

The standard form of a quadratic imparts significant information about the equation:

• a determines the direction of the parabola (concavity).
• b influences the location of the vertex horizontally.
• c represents the y-intercept of the graph.

Identifying the values of these coefficients allows us to understand the behavior of the quadratic model and its graph.

Polynomials

In algebra, polynomials are expressions consisting of variables (also known as indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A quadratic equation is a specific type of polynomial with a degree of two, which means the highest power of the variable is two.

Polynomials are classified based on the number of terms they have and their degree. Here is the general structure:

• Monomials have one term (e.g., 3x^2).
• Binomials have two terms (e.g., 3x^2 - 2x).
• Trinomials have three terms (e.g., 3x^2 - 2x + 4).

In the context of the exercise, f(x) = 4x^2 is a monomial because it consists of a single term and it is a polynomial of degree two - making it a quadratic polynomial. Understanding the nature of polynomials is essential in algebra as they appear frequently in various forms and applications.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The purpose of an algebraic expression is to represent a particular mathematical idea or quantity.

Unlike equations, algebraic expressions do not have an equal sign; they are not statements of equality but rather terms grouped together in a meaningful way that can be used within equations. For instance, in the quadratic model f(x) = 4x^2, the right-hand side 4x^2 is an algebraic expression representing the model's value given a specific input x.

Algebraic expressions can be simplified, factored, expanded, and manipulated in various ways to solve problems or to express the problems in a more workable format. A profound understating of algebraic expressions is vital for students because it is the foundation upon which most of algebra, and by extension calculus and other higher mathematics, are built.