Problem 37 - Algebra for College Students

Question

Is \(f(x)=\left(3 x^{2}-2\right)-(2 x+1)\) a quadratic function? Explain your answer.

Step-by-Step Solution

Answer

Yes, it is a quadratic function because it simplifies to \(3x^2 - 2x - 3\), with \(a = 3 \neq 0\).

Step 1: Understand the Problem

To determine if the function \(f(x) = (3x^2 - 2) - (2x + 1)\) is quadratic, we need to express it in the standard form of a quadratic function, which is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).

Step 2: Simplify the Expression

Start by expanding the given function: \(f(x) = 3x^2 - 2 - 2x - 1\). Now, combine like terms to simplify the expression.

Step 3: Combine Like Terms

Combine the constants and the linear terms: \(f(x) = 3x^2 - 2x - 3\). This combines the \(-2\) and \(-1\) to get \(-3\).

Step 4: Identify the Standard Form

The simplified expression is \(f(x) = 3x^2 - 2x - 3\). This fits the standard form of a quadratic function \(ax^2 + bx + c\), where \(a = 3\), \(b = -2\), and \(c = -3\).

Step 5: Check the Quadratic Term

Verify that the coefficient \(a\) of the quadratic term \(x^2\) is not zero. As long as \(a eq 0\), the function is considered a quadratic. Here, \(a = 3\), which is not zero.

Key Concepts

Standard Form of Quadratic Equation

Understanding the standard form of a quadratic equation is crucial. This form is expressed as \( ax^2 + bx + c \). It provides a framework to easily identify the behavior and characteristics of quadratic functions. Here, \( a \), \( b \), and \( c \) represent real number coefficients. The significance of these components are:

• \( a \) is the coefficient of the \( x^2 \) term. This value is crucial because it determines the parabola's direction—upward when \( a \) is positive, and downward when negative. Additionally, \( a \) must be non-zero for the equation to truly be quadratic.


• \( b \) represents the coefficient of the linear or \( x \) term, impacting the slope and y-intercept.


• \( c \) is the constant term and determines where the parabola crosses the y-axis.

By converting expressions into this form, it becomes straightforward to analyze and work with quadratic functions.

Polynomial Simplification

Simplifying polynomials is key to tackling algebraic expressions effectively. The original expression \((3x^2 - 2) - (2x + 1)\) requires expanding and combining terms in a straightforward manner. Start with:

• Expand the subexpressions: Distribute the subtraction across the brackets which gives \( 3x^2 - 2 - 2x - 1 \).


• Combine like terms: Group terms based on their degree. Here, \( 3x^2 \) stays as is, while the constants \(-2\) and \(-1\) add to \(-3\), and \(-2x\) stands alone as the linear term.

Simplification reduces the complexity of the function, resulting in \( 3x^2 - 2x - 3 \). This process aids in seeing the structure of quadratic equations more clearly and allows for easy comparison to the standard form.

Coefficient Identification

Identifying coefficients in expressions is essential for understanding polynomials' structure. In a quadratic function \( ax^2 + bx + c \), each term's coefficient tells a story about the function's graph and solutions. Consider:

• Coefficient \( a = 3 \): As the leading coefficient, it confirms the parabola is opening upwards and maintains the function's quadratic nature.


• Coefficient \( b = -2 \): This impacts the symmetry and slope of the parabola.


• Constant \( c = -3 \): Demonstrates where the graph intercepts the y-axis.

By pinpointing these values, you gain insights into graph features and solutions—important for graphing, solving, and interpreting quadratic equations.