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A quadratic function is a type of polynomial that is essential in both algebra and geometry. Its general form is \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The most basic characteristic of a quadratic function is that it has a variable \(x\) raised to the second power.

• Quadratic functions graph as parabolas, which can open upwards or downwards depending on the coefficient \(a\).
• The vertex of the parabola is the point at which it changes direction.
• The line of symmetry runs vertically through the vertex.

In the given exercise, the function is \(f(x) = x^2 + 1\). This function doesn't have the \(bx\) term, making it a simpler case of a quadratic function. It opens upwards because the coefficient of \(x^2\) is positive (\(+1\)), and its vertex is at \((0, 1)\). Evaluating this function at different values of \(x\) involves substituting these values into the expression and solving.

The substitution method is an essential technique in algebra used for evaluating functions and solving equations. It involves replacing the variable with a specific numerical value to simplify or solve an expression.

• Identify the function you need to evaluate. For example, in the exercise, this is \(f(x) = x^2 + 1\).
• Determine the value to substitute for \(x\) – in this exercise, these values are 1, -2, and 3.
• Replace \(x\) with the chosen value and calculate e.g., substituting 1, the expression turns into \((1)^2 + 1 = 2\).

It's simple yet powerful, allowing you to easily find specific output values of a function by substituting different inputs.

Algebraic expressions are fundamental in mathematics and involve numbers, variables, and operations. Simplifying these expressions ensures clarity and helps solve problems effectively. In the given function \(f(x) = x^2 + 1\), we focus on computing the simplest form by evaluating each specific input. The simplification process is as follows:

• Substitute \(x\) with a given number.
• Perform the operations following the standard order: exponents first, then addition.
• For \(f(1)\), it is \((1)^2 + 1 = 2\).
• For \(f(-2)\), it becomes \((-2)^2 + 1 = 5\).
• For \(f(3)\), simplifying gives \((3)^2 + 1 = 10\).

Using these steps, you transform complex expressions into simple numerical results, making it easier to understand and interpret the behavior of algebraic functions.