Problem 13 - Algebra for College Students

Question

If \(f(x)=-x^{2}+3 x+5\), find \(f(-a), f(a+6)\), and \(f(-a+1)\).

Step-by-Step Solution

Answer

\(f(-a) = -a^{2} - 3a + 5\), \(f(a+6) = -a^{2} - 9a - 13\), \(f(-a+1) = -a^{2} - a + 7\).

Step 1: Understanding the Function

The function given is a quadratic function: \(f(x) = -x^{2} + 3x + 5\). We will substitute the given values into this function to find \(f(-a)\), \(f(a+6)\), and \(f(-a+1)\).

Step 2: Finding \(f(-a)\)

Substitute \(-a\) into the function: \(f(-a) = -(-a)^{2} + 3(-a) + 5\). Simplify the terms: \(f(-a) = -a^{2} - 3a + 5\).

Step 3: Finding \(f(a+6)\)

Substitute \(a+6\) into the function: \(f(a+6) = -(a+6)^{2} + 3(a+6) + 5\). First, expand \((a+6)^{2}\) to \(a^{2} + 12a + 36\), then substitute and simplify: \(f(a+6) = -a^{2} - 12a - 36 + 3a + 18 + 5\). Therefore, \(f(a+6) = -a^{2} - 9a - 13\).

Step 4: Finding \(f(-a+1)\)

Substitute \(-a+1\) into the function: \(f(-a+1) = -(-a+1)^{2} + 3(-a+1) + 5\). Expand \((-a+1)^{2}\) to \(a^{2} - 2a + 1\), then substitute and simplify: \(f(-a+1) = -a^{2} + 2a - 1 - 3a + 3 + 5\). Therefore, \(f(-a+1) = -a^{2} - a + 7\).

Key Concepts

Polynomials and Their Expressions

Polynomial expressions are mathematical phrases that involve variables and coefficients. The terms in these expressions consist of a variable raised to a power, and each variable may have different coefficients. For instance, the expression \(-x^2 + 3x + 5\) is a polynomial. It includes the terms \(-x^2\), \(3x\), and \(5\).
Polynomials are important in algebra because they can describe a wide range of numbers and functions. For example, a quadratic function, like the one in our exercise, is a polynomial of degree 2, implying the highest power of the variable \(x\) is squared. Understanding polynomial expressions is crucial because they form the foundation for solving various types of equations. Recognizing these terms will help you manipulate and simplify expressions efficiently.

Understanding Function Evaluation

Function evaluation involves substituting specific values into a function to calculate an output. It's like a machine that takes an input value, performs calculations, and produces an output.
When asked to find values like \(f(-a)\), \(f(a+6)\), and \(f(-a+1)\), for example, it means replacing \(x\) with the given expressions. After replacing, you simplify the result using algebraic operations.

• Substitute the entire replacement for \(x\) in the function.
• Follow the order of operations: parentheses, exponents, multiplication/division, and addition/subtraction.
• Simplify the expression accordingly.

For instance, to evaluate \(f(-a)\), you substitute \(-a\) for every occurrence of \(x\) in \(-x^2 + 3x + 5\). This results in the polynomial \(-a^2 - 3a + 5\). Consistently following these steps ensures accuracy.

The Art of Simplifying Expressions

Simplifying expressions is a key skill in algebra, making complex expressions more manageable. It involves combining like terms and reducing parts of the expression wherever possible.
When simplifying, observe if terms have the same variable raised to the same power. These are like terms and can be combined.

• Perform operations like expanding: turn \((a+6)^2\) into\(a^2 + 12a + 36\).
• Next, distribute any coefficients across terms.
• Combine like terms to simplify: for \(f(a+6)\), the result turns into\(-a^2 - 9a - 13\).

Remember that simplification doesn't change the value but provides a cleaner and quicker way to work with algebraic expressions. It's all about making problems easier to solve while retaining the original expression's value.