Problem 52 - Intermediate Algebra

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: Understand the Function

The function given is a quadratic function in the form of \(f(x) = -x^2 + 3x + 5\). This function takes any input \(x\) and gives an output by replacing \(x\) with the input value.

Step 2: Evaluate f(-a)

To find \(f(-a)\), replace every instance of \(x\) in the function with \(-a\):\[f(-a) = -(-a)^2 + 3(-a) + 5\]Calculate each term:\[f(-a) = -(a^2) - 3a + 5\]Thus, \(f(-a) = -a^2 - 3a + 5\).

Step 3: Evaluate f(a+6)

To find \(f(a+6)\), replace \(x\) with \(a+6\) in the function:\[f(a+6) = -(a+6)^2 + 3(a+6) + 5\]Expand and simplify each part:\[= -(a^2 + 12a + 36) + 3a + 18 + 5\]Combine like terms:\[= -a^2 - 12a - 36 + 3a + 23\]\[f(a+6) = -a^2 - 9a - 13\]

Step 4: Evaluate f(-a+1)

To find \(f(-a+1)\), replace \(x\) with \(-a+1\):\[f(-a+1) = -(-a+1)^2 + 3(-a+1) + 5\]Expand and simplify:\[= -(a^2 - 2a + 1) - 3a + 3 + 5\]Combine like terms:\[= -a^2 + 2a - 1 - 3a + 8\]\[f(-a+1) = -a^2 - a + 7\]

Key Concepts

Function Evaluation

Function evaluation is akin to placing values into an equation to see what results we get. In the exercise, we were given a quadratic function: \(f(x) = -x^2 + 3x + 5\).
To evaluate the function, we substitute the specified value for \(x\) and calculate the result. For instance:

• To find \(f(-a)\), substitute \(-a\) for every occurrence of \(x\); solve the expression to get \(-a^2 - 3a + 5\).
• For \(f(a+6)\), we substitute \(a+6\) into the function and simplify, arriving at \(-a^2 - 9a - 13\).
• Similarly, substituting \(-a+1\) into \(f(x)\) will give us \(-a^2 - a + 7\).

Understanding how to replace \(x\) with different inputs and simplifying the expressions helps in deriving the function's value for specific inputs.

Algebraic Expressions

Algebraic expressions represent numbers using symbols and variable operations. They are crucial for managing variable inputs in equations. In the function \(-x^2 + 3x + 5\), each term involves algebraic manipulation.
When evaluating expressions, consider operations like:

• Squaring terms: For instance, \((-a)^2\) becomes \(a^2\).
• Substituting and combining terms: Replace variables like \(x\) in the function with expressions (e.g., \(-a\), \(a+6\)) to simplify and evaluate results.
• Distributing terms: Utilize distribution to expand terms, such as \(-(a^2 + 12a + 36)\), turning it into \(-a^2 - 12a - 36\).

Mastering these algebraic operations makes it easier to substitute and evaluate expressions within functions.

Polynomials

A polynomial is a mathematical expression consisting of variables raised to different powers, added or subtracted together. In our function, \(-x^2 + 3x + 5\), the expression involves:

• A quadratic term: \(-x^2\), which is a typical feature in a polynomial.
• Linear terms: Terms like \(3x\) with variable powers of one.
• Constant terms: Numbers like \(+5\), independent of variables.

When working with polynomials:

• Identify each term and its degree (highest power of \(x\) in the function).
• Understand that polynomials can describe curved shapes; this function, in particular, is a parabola due to its quadratic nature.
• Simplifying polynomials when evaluating them allows us to analyze and manipulate their behavior more effectively.

The interplay of these terms in equations like our quadratic function involves substituting values and refining expressions to investigate polynomial characteristics.