Problem 6 - Algebra A Combined Function

Question

If \(P(x)=x^{2}+x+1\) and \(Q(x)=5 x^{2}-1,\) find each function value. $$ Q(0) $$

Step-by-Step Solution

Answer

Q(0) = -1.

Step 1: Substitute into the function

To find the value of \(Q(0)\), substitute \(0\) for \(x\) in the function \(Q(x)=5x^2-1\).

Step 2: Simplify the expression

Once you substitute \(0\) into \(5x^2-1\), you get \(5(0)^2 - 1\). This simplifies to \(0 - 1\).

Step 3: Calculate the result

After simplifying the expression, you have \(0 - 1 = -1\). Hence, \(Q(0) = -1\).

Key Concepts

Quadratic Functions

In mathematics, a quadratic function is a polynomial function of the form \(f(x) = ax^2 + bx + c\). It is called quadratic because "quad" means square, indicating that the highest power of the variable \(x\) is 2. This makes a distinctive U-shaped curve called a parabola when graphed.
Understanding quadratic functions is crucial in algebra because they frequently occur in a range of real-world contexts, such as physics, engineering, and economics.

• Standard Form: The standard form of a quadratic function is expressed as \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
• Vertex Form: Quadratic functions can also be represented in vertex form as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
• Factored Form: Finally, the factored form is \(a(x-p)(x-q)\), where \(p\) and \(q\) are the roots of the function.

By examining different forms, you can gain insights into the properties of the parabola, such as its intercepts, vertex, and direction. Understanding these properties helps solve equations and contextual problems efficiently.

Substitution Method

The substitution method is a key strategy in algebra for solving equations or simplifying expressions. This method involves replacing a variable with a given value or another expression.
For example, in the original exercise, finding \(Q(0)\) requires substituting \(0\) for \(x\) in the equation \(Q(x) = 5x^2 - 1\). Here's how it works step by step:

• Identify the variable: Determine which variable to substitute. In this case, it is \(x\).
• Perform the substitution: Replace each instance of \(x\) with \(0\) to get \(5(0)^2 - 1\).
• Simplify the expression: Simplify the resulting expression. Here, \(5(0)^2 - 1 = 0 - 1 = -1\).

This technique is vital in evaluating functions because it facilitates finding function values, such as calculating a specific output or expression's result with a given input.

Polynomials

A polynomial is an expression made up of variables, coefficients, and non-negative integer exponents of variables. Polynomials can be simple, with only a single term called a monomial, or complex, with many terms such as binomials (two terms) or trinomials (three terms). Each polynomial type helps express a broader range of equations and functions.

• General Form: A polynomial in one variable \(x\) is in the form \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n\) is the leading coefficient and \(a_0\) is the constant term.
• Degree: The degree of the polynomial is the highest power of the variable. For example, in \(Q(x) = 5x^2 - 1\), the degree is 2.

Polynomials are crucial in algebra as they form the foundation for more complex mathematical concepts, including polynomial equations, factorizations, and polynomial functions. Mastering polynomials equips students with the skills to tackle a variety of mathematical challenges.