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In mathematics, the evaluation of a polynomial function involves plugging specific values into the function to determine its result. When you encounter a problem that requires finding several evaluations, like evaluating a function for different values of \(x\), it's referred to as function evaluation. This process helps in understanding how a given function behaves with different inputs.
The polynomial function in this exercise is \(f(x)=4x^2-3x+2\). To evaluate a function, we simply replace the variable \(x\) with the desired value and follow through with the arithmetic operations:

• For \(f(0)\), substitute \(x = 0\), leading to \(f(0) = 4(0)^2 - 3(0) + 2 = 2\).
• For \(f(3)\), substitute \(x = 3\), leading to \(f(3) = 4(3)^2 - 3(3) + 2 = 29\).
• For \(f(-1)\), substitute \(x = -1\), leading to \(f(-1) = 4(-1)^2 - 3(-1) + 2 = 9\).

Each evaluation ultimately helps to find specific outputs at certain inputs. These specific points can give insight into the pattern or behavior of the function.

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition or multiplication. In the provided exercise, the expression \(4x^2 - 3x + 2\) represents a polynomial function of \(x\). Each component of this expression has a distinct role:

• The term \(4x^2\) is called a quadratic term, since the variable \(x\) is raised to the power of 2.
• \(-3x\) is referred to as the linear term due to the presence of the variable \(x\) to the power of 1.
• Lastly, \(+2\) represents the constant term, which stays the same regardless of the value of \(x\).

Algebraic expressions, specifically polynomials, are used to model various real-world situations. They provide a convenient and standardized way to represent relationships where one quantity depends on another.

The substitution method, also known as plugging in values, is a fundamental technique in algebra used to evaluate expressions or solve equations. It's simply the action of replacing a variable with a numerical value or expression. This method is invaluable in simplifying and calculating the result of an expression.
Consider our polynomial \(f(x) = 4x^2 - 3x + 2\). To evaluate \(f(x)\) for specific values, we employ substitution by replacing \(x\) with numbers:

• Place \(x = 0\) in \(f(x)\) to find \(f(0) = 2\).
• Insert \(x = 3\) to compute \(f(3) = 29\).
• Replace \(x = -1\) to get \(f(-1) = 9\).

The substitution method is not limited to evaluating functions but also plays a crucial role in solving equations. In exercises that involve multiple values like ours, it becomes especially powerful, allowing for quick calculations and deeper insights into functional behavior.