Problem 96 - Algebra 2

Question

PREREQUISITE SKILL. Evaluate each function for the given value. $$ f(x)=3 x^{2}+7 x, x=-2 $$

Step-by-Step Solution

Answer

The value of the function at \( x = -2 \) is \( -2 \).

Step 1: Substitute the Value of x

First, substitute the value of \( x = -2 \) into the function \( f(x) = 3x^2 + 7x \). So, we replace \( x \) with \(-2\) in the expression.\[ f(-2) = 3(-2)^2 + 7(-2) \]

Step 2: Evaluate the Expression Inside the Parentheses

Calculate the value of \((-2)^2\). Raising \(-2\) to the 2nd power means multiplying \(-2\) by itself.\[ (-2)^2 = 4 \]

Step 3: Multiply by the Coefficient

Now, multiply the result from Step 2 by the coefficient 3. This is part of the expression for \( f(-2) \).\[ 3 imes 4 = 12 \]

Step 4: Evaluate the Linear Term

Calculate the result of the linear term by multiplying 7 by \(-2\).\[ 7 imes (-2) = -14 \]

Step 5: Combine the Results

Finally, add the results from Step 3 and Step 4 together to find \( f(-2) \).\[ f(-2) = 12 + (-14) = -2 \]

Key Concepts

Substitution Method

The substitution method is a straightforward technique used for evaluating functions by replacing variables with given values. This approach allows us to find the output of a function at specific points. In our example, we have the function \(f(x) = 3x^2 + 7x\) and need to determine the value of the function when \(x = -2\).

To apply the substitution method, we start by replacing every instance of the variable \(x\) in the function with the given value, which in this case is \(-2\). Thus, the function becomes \(f(-2) = 3(-2)^2 + 7(-2)\). The substitution method simplifies the process of evaluating functions, making it ideal for both polynomial and more complex functions. By seamlessly transforming a function into a numeric expression, we can calculate its value quickly and accurately.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of variables, each multiplied by a coefficient. A general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a\) is a coefficient and \(n\) is a non-negative integer.

In our exercise, the function \(f(x) = 3x^2 + 7x\) is a polynomial of degree 2, or a quadratic polynomial, because the highest power of \(x\) is 2. Each term in a polynomial function is composed of a variable, a coefficient, and an exponent.

Understanding polynomial functions is crucial because they often model real-world phenomena and appear in various subjects, from physics to economics. Recognizing the forms and behaviors of different polynomial functions can help with simplifying expressions, solving equations, and graphing.

Algebraic Operations

Algebraic operations involve the manipulation of algebraic expressions using basic arithmetic operations such as addition, subtraction, multiplication, and division. When evaluating functions like \(f(x) = 3x^2 + 7x\), these operations enable us to transform variable expressions into concrete numbers.

The steps to solve our problem include several basic algebraic operations, such as:

• Calculating the power of a number: We raised \(-2\) to the power of 2, resulting in 4.
• Multiplying: We multiplied 3 by 4 to process the quadratic term and 7 by \(-2\) for the linear term.
• Adding results: Finally, we combined 12 and \(-14\) through addition to get \(f(-2) = -2\).

Using algebraic operations, we can take a symbolic representation of a function, perform calculations, and derive a numerical result. Understanding and mastering these operations is foundational in mathematics and aids in problem-solving across various disciplines.