Problem 43 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-3 x+2\) $$ f(-3) $$

Step-by-Step Solution

Answer

The value of \(f(-3)\) is 20.

Step 1: Identify the Function and Input Value

We are given the function \(f(x) = x^2 - 3x + 2\), and we need to find \(f(-3)\). This means we will substitute \(-3\) for \(x\) in the function.

Step 2: Substitute \(-3\) into the Function

Replace every instance of \(x\) in the function \(f(x) = x^2 - 3x + 2\) with \(-3\). We get:\[f(-3) = (-3)^2 - 3(-3) + 2\]

Step 3: Simplify the Expression

Calculate each term:- \((-3)^2 = 9\)- \(-3(-3) = 9\)- \(+ 2 = 2\)So, the expression is simplified as:\[f(-3) = 9 + 9 + 2\]

Step 4: Add the Results Together

Add the simplified results: \(9 + 9 + 2 = 20\).

Step 5: Conclude the Calculation

Thus, the value of \(f(-3)\) is \(20\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number exponents and their coefficients. The general form of a polynomial function of degree \(n\) is:

• \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\)

In this formula:

• \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants known as coefficients.
• \(n\) is the highest power of \(x\), and it is considered the degree of the polynomial.
• The exploration often focuses on the behavior and graph of the function.

Understanding polynomial functions is fundamental in algebra and calculus. They allow you to create and analyze curves and surfaces in mathematics.
In our exercise, the polynomial function given is \(f(x) = x^2 - 3x + 2\), a quadratic polynomial since the highest power of \(x\) is \(2\). Polynomials like this are common and include various components that can impact their shape and symmetry on a graph. Their roots, axis of symmetry, and vertex are essential characteristics when graphed.

Substitution Method

The substitution method is a powerful algebraic tool used to solve equations or plug specific values into functions. This technique involves replacing variables in the function with particular numbers or other expressions. It's particularly useful in function evaluation and equation solving because it provides a straightforward way to find exact values.
Here's how substitution works in the context of our exercise:

• You start with the function \(f(x) = x^2 - 3x + 2\).
• You need to find \(f(-3)\), which involves substituting \(-3\) into the function where \(x\) appears.

This method allows you to convert the variable-based expression directly into a numerical one, making calculations more manageable.

Function Evaluation

Function evaluation is the process of determining the output of a function given an input value. This involves substituting a number into the variable of a function and then simplifying the expression to reach a result.
Evaluating functions plays a crucial role in understanding their behavior at specific points. For our given quadratic function \(f(x) = x^2 - 3x + 2\), the task is to evaluate \(f(-3)\).

• First, substitute \(-3\) into the function: \(f(-3) = (-3)^2 - 3(-3) + 2\).
• Calculate each term: \((-3)^2 = 9\), \(-3(-3) = 9\), and \(+ 2 = 2\).
• Add all the results: \(9 + 9 + 2 = 20\).

After performing these steps, it is clear that the value of the function at \(x = -3\) is \(20\). This illustrates how the function behaves at specific points and aids in sketching the graph of the function, as understanding its value at various points can help determine its shape.