Problem 53 - Algebra 2

Question

Given \(f(x)=x^{2}-5 x+6,\) find each value. $$ f(-2) $$

Step-by-Step Solution

Answer

The value of \(f(-2)\) is 20.

Step 1: Identify the given function

The function provided is a quadratic function, written as \[f(x) = x^2 - 5x + 6\]where \(x\) is the input variable.

Step 2: Substitute the given value into the function

We are asked to evaluate the function for \(x = -2\). Substitute \(-2\) for \(x\) within the function:\[f(-2) = (-2)^2 - 5(-2) + 6\]

Step 3: Simplify the function

First, calculate \((-2)^2\): \[(-2)^2 = 4\]Next, calculate \(-5(-2)\):\[-5(-2) = 10\]Now substitute these values back into the function:\[f(-2) = 4 + 10 + 6\]

Step 4: Perform the final calculation

Add the values to complete the function evaluation:\[f(-2) = 4 + 10 + 6 = 20\]

Step 5: State the final answer

Thus, the value of the function \(f(x)\) at \(x = -2\) is 20.

Key Concepts

Function Evaluation

When we talk about function evaluation, we're referring to finding the output of a function for a specific input value. Consider a function as a kind of machine that takes input, processes it according to certain rules, and produces an output. In this case, our machine is the quadratic function given by \( f(x) = x^2 - 5x + 6 \). To evaluate a function like \( f(x) \), we simply input a specific value for \( x \), and the machine will carry out the operations indicated by the function's formula. Let's walk through the steps:

• Take the given input value, which is \( x = -2 \).
• Substitute \( -2 \) into the function wherever you see \( x \).
• Complete any necessary calculations to find the result.

By following these steps, you effectively determine "what comes out" of the function when a particular "input" enters.

Polynomial Functions

Polynomial functions are like a special club of functions. They're expressions made from variables, coefficients, and the potent operations of addition, subtraction, and multiplication. What makes them unique? It's their degree. The degree of a polynomial is the highest power of the variable present in its expression. For instance, a quadratic function, like our example \( f(x) = x^2 - 5x + 6 \), is a polynomial of degree 2. The 2 here comes from the highest power, \( x^2 \), which dictates the function's behavior. Quadratic functions tend to produce parabolic graph shapes, which are symmetrical.

• Term: Each component (like \(x^2\)) is known as a term.
• Coefficient: The number in front of the variable (like \(-5\) in \(-5x\)) is its coefficient.
• Constant: A term without variable (like \(+6\)) is constant.

Learning about these elements helps simplify polynomials and understand their varied and fascinating applications.

Substitution Method

The substitution method is an essential tool in mathematics, used in evaluating functions and solving equations. It simplifies the process by replacing variables with known values. This approach leads to easier calculations and often provides solutions in fewer steps. When applying the substitution method, consider the following:

• Identify the variable you want to substitute. In our example, \( x \) was the target.
• Replace this variable with a given value, such as \( x = -2 \).
• Execute standard arithmetic operations with accuracy and care.

This technique is powerful in breaking down complex expressions into more manageable calculations. By doing so, it allows even intricate-looking problems to be approached with confidence and precision, leading to correct and efficient outcomes.