Problem 95 - Algebra 2

Question

PREREQUISITE SKILL. Evaluate each function for the given value. $$ f(x)=-x^{2}-4 x+5, x=-3 $$

Step-by-Step Solution

Answer

The value of the function at \(x = -3\) is 8.

Step 1: Substitute the Value of x

First, identify the given function and the value of \(x\) that we need to substitute into the function. The function is \(f(x) = -x^2 - 4x + 5\), and the given value of \(x\) is \(-3\). Substitute \(-3\) for \(x\) in the function to get \(f(-3) = -(-3)^2 - 4(-3) + 5\).

Step 2: Simplify the Squared Term

Evaluate the squared term \((-3)^2\). We have \((-3)^2 = 9\). Substitute this back into the expression: \(-9 - 4(-3) + 5\).

Step 3: Simplify the Multiplication

Continue by evaluating the multiplication term \(-4(-3)\). This results in \(12\) because multiplying two negative numbers yields a positive product. Now substitute 12 into the expression: \(-9 + 12 + 5\).

Step 4: Perform the Addition and Subtraction

Finally, perform the operations from left to right: \(-9 + 12 = 3\), and then \(3 + 5 = 8\). Therefore, \(f(-3) = 8\).

Key Concepts

Understanding Polynomial Functions

A polynomial function is a type of mathematical function that includes terms composed of variables, exponents, and constants. The general form of a polynomial is given by

• \( a_nx^n + a_{n-1}x^{n-1} + \, \ldots \, + a_1x + a_0 \)

where \(a_n, a_{n-1}, \ldots, a_1, a_0\) are constants, and \(n\) is a non-negative integer.

These functions can be both simple, like linear functions, or more complex with higher powers and multiple terms. In our example, the function \(f(x) = -x^2 - 4x + 5\) is a quadratic polynomial, meaning it includes an \(x^2\) term.

Quadratic polynomials are key in algebra because they can demonstrate a range of behaviors such as the direction of a parabola opening, vertex positions, and more. Let's explore further how to evaluate such functions with substitution.

The Substitution Method Made Easy

The substitution method in function evaluation is quite straightforward. It involves replacing the variable in the polynomial function with a specific value to find the function’s resulting value.

Here is how it works in simple steps:

• Identify the function and the value of \(x\): For example, in \(f(x) = -x^2 - 4x + 5\), we are instructed to evaluate for \(x = -3\).
• Replace \(x\) with the given value: Substitute \(-3\) into the function, changing \(f(x)\) to \(f(-3)\).
• Perform arithmetic operations: Simplify the expression carefully, starting with exponents, followed by multiplication or division, and then addition or subtraction.

The beauty of this method lies in its systematic approach. It is consistent and can be applied to any polynomial function to efficiently find its value at any point.

Working With Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

In our problem, we’re working with the expression \(-x^2 - 4x + 5\). An important skill in handling algebraic expressions is simplifying them while maintaining accuracy.

• Square Calculation: Evaluate exponents first. For example, \((-3)^2\) equals \(9\).
• Addressing Terms with Negative Signs: Always apply arithmetic rules correctly, such as understanding multiplication rules for negative numbers.
• Simplification: Simplify the expression step-by-step; it helps in keeping track of different components and ensuring correct calculations. For this, after simplification, \(-9 + 12 + 5\) equals \(8\).

Efficiently working through algebraic expressions can be greatly enhanced by practicing these steps carefully, enabling you to solve functions accurately with ease.