Problem 58 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-7 x+5\) \(f(2)\)

Step-by-Step Solution

Answer

-5

Step 1: Substitute the value into the function

We need to find the value of the function \(f(x)\) at \(x=2\). Start by replacing \(x\) with \(2\) in the function formula. Thus, \(f(2) = 2^2 - 7 \times 2 + 5\).

Step 2: Simplify the expression

Calculate the power and multiplications in the expression. First, calculate \(2^2\), which is \(4\), and \(-7 \times 2\), which is \(-14\). The expression becomes \(4 - 14 + 5\).

Step 3: Perform the addition and subtraction

Now, simplify \(4 - 14 + 5\). First, perform \(4 - 14\) to get \(-10\). Then add \(-10 + 5\) to get \(-5\).

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables and coefficients combined using addition, subtraction, and multiplication. They have terms featuring variables raised to a whole number power. For example, a polynomial might look like this: \(f(x) = ax^n + bx^{n-1} + \ldots + zx^{0}\). In this equation:

• \(a, b, \ldots, z\) are the coefficients
• \(x\) is the variable
• \(n\) and other exponents are whole numbers

Polynomials can be classified by degree, which is the highest power of the variable in the expression. A polynomial's degree helps us understand its general behavior and graph shape. The example \(f(x) = x^2 - 7x + 5\) is a quadratic polynomial because its highest degree is 2.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. This means replacing the variable in the function with a given number. For the polynomial \(f(x) = x^2 - 7x + 5\), evaluating it at \(x = 2\) involves:

• Substituting \(2\) in place of \(x\) in the expression
• Calculating the powers and multiplications first
• Finally, performing the addition and subtraction

By doing this, you transform the abstract function into a concrete number. In our example, calculating each term step-by-step helps ensure we don't miss any arithmetic details. This process is relatively simple but crucial for understanding how changing the input affects the output in function analysis.

Quadratic Expressions

Quadratic expressions are a specific type of polynomial expression where the highest degree of the variable is 2. These expressions can be thought of as having a 'U' shaped graph, known as a parabola. A general quadratic form looks like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

• \(a\) determines the opening direction (upward if positive, downward if negative)
• \(b\) affects the parabola's symmetry
• \(c\) provides the y-intercept, or the point where the graph crosses the y-axis

Our function, \(f(x) = x^2 - 7x + 5\), fits this form. When simplifying such functions, focusing on each term systematically is key. Handling quadratic expressions frequently helps us understand their properties and solve related equations efficiently.