Problem 60 - Algebra 2

Question

Find each value if \(f(x)=x^{2}-7 x+5\) \(f(-3)\)

Step-by-Step Solution

Answer

The value of \( f(-3) \) is 35.

Step 1: Understand the Function

The given function is \( f(x) = x^2 - 7x + 5 \). We will evaluate this function at a specific point, in this case, \( x = -3 \).

Step 2: Substitute the Value into the Function

Replace \( x \) with \( -3 \) in the function. This gives us:\[ f(-3) = (-3)^2 - 7(-3) + 5 \]

Step 3: Calculate the Squared Term

Calculate \((-3)^2\), which equals 9. Plug this back into the function, giving:\[ f(-3) = 9 - 7(-3) + 5 \]

Step 4: Calculate the Product

Calculate \(-7 \times (-3)\), which equals 21. Substitute this into the equation:\[ f(-3) = 9 + 21 + 5 \]

Step 5: Perform the Addition

Finally, add the numbers together: 9 + 21 + 5 equals 35. So, the value of the function at \( x = -3 \) is \( 35 \).

Key Concepts

Understanding Function Evaluation

Function evaluation is key to understanding how inputs affect a function's output. In mathematical terms, a function like \( f(x) \) is a rule that assigns an output to each input \( x \). The function \( f(x) = x^2 - 7x + 5 \) is a classic quadratic function. Evaluating this involves substituting a specific value for \( x \), such as \( x = -3 \). This means you replace every \( x \) in the function with \(-3\). The calculation then follows the standard order of operations:

• Compute any exponents or powers, such as \((-3)^2\).
• Perform multiplication, such as \(-7\times(-3)\).
• Execute addition or subtraction, completing the evaluation.

This step-by-step approach ensures accurate function evaluation, transforming a potentially complex expression into a straightforward numerical result.

Exploring Polynomials

Polynomials are algebraic expressions that consist of variables, coefficients, and exponents which are combined using addition, subtraction, and multiplication (but not division by a variable). In the function \(f(x) = x^2 - 7x + 5\), we have a quadratic polynomial characterized by the highest exponent of 2. Quadratic polynomials are fundamental in algebra for:

• Modeling real-world situations, such as projectile motion.
• Finding maximum or minimum values through vertex analysis.
• Solving quadratic equations which have numerous applications in geometry and physics.

Understanding the structure of a polynomial, such as its degree (highest power) and terms, helps in simplifying and evaluating expressions. Identifying coefficients like 1 in \(x^2\), \(-7\) in \(-7x\), and constant terms, like 5, define the polynomial's behavior and graph shape.

Algebraic Expressions Demystified

Algebraic expressions like \(x^2 - 7x + 5\) help us generalize mathematical ideas and relationships. These expressions combine constants, variables, and operators to form meaningful relationships that can be solved or evaluated. Key parts of these expressions include:

• Terms: Components separated by addition or subtraction (e.g., \(x^2, -7x, \) and \(+5\)).
• Coefficients: Numerical factors that multiply the variable parts (e.g., \(1\) in \(x^2\) and \(-7\) in \(-7x\)).
• Constants: Standalone numbers without variables (e.g., \(+5\)).

Mastering algebraic expressions allows you to manipulate equations and solve for unknowns. This is fundamental in finding solutions to algebraic equations, such as when evaluating functions or solving for roots in polynomials. Making these concepts accessible involves practice and familiarity with each component, leading to the ability to see patterns and solutions quickly.