Problem 38 - Algebra 2

Question

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=\frac{1}{2} x^{4}-2 x^{2}+4\)

Step-by-Step Solution

Answer

\(p(4) = 100\), \(p(-2) = 4\).

Step 1: Plug in 4 for x

Let's find \(p(4)\). Substitute 4 for \(x\) in the polynomial function: \[ p(4) = \frac{1}{2}(4)^4 - 2(4)^2 + 4 \]

Step 2: Calculate each term for p(4)

Calculate each term separately:1. \(\frac{1}{2}(4)^4 = \frac{1}{2} \times 256 = 128\)2. \(2(4)^2 = 2 \times 16 = 32\)3. Constant term = 4 Therefore, \( p(4) = 128 - 32 + 4 \).

Step 3: Simplify p(4)

Simplify the equation from Step 2:\[ p(4) = 128 - 32 + 4 = 100 \]

Step 4: Plug in -2 for x

Now, let's find \(p(-2)\). Substitute \(-2\) for \(x\) in the polynomial function: \[ p(-2) = \frac{1}{2}(-2)^4 - 2(-2)^2 + 4 \]

Step 5: Calculate each term for p(-2)

Calculate each term separately:1. \(\frac{1}{2}(-2)^4 = \frac{1}{2} \times 16 = 8\)2. \(2(-2)^2 = 2 \times 4 = 8\)3. Constant term = 4 Therefore, \( p(-2) = 8 - 8 + 4 \).

Step 6: Simplify p(-2)

Simplify the equation from Step 5:\[ p(-2) = 8 - 8 + 4 = 4 \]

Key Concepts

Function Evaluation

Function evaluation is the process of determining the value of a function for specific input values. In simple terms, it's like finding an answer to a problem by inserting a particular number into the function's formula. This is crucial in understanding how the function behaves with different inputs.
To evaluate a function, you take a given function, such as the polynomial function, and replace the variable with a specified value.
To illustrate, let's look at the polynomial function given in the exercise:

• For the function \( p(x) = \frac{1}{2}x^4 - 2x^2 + 4 \), evaluating \( p(4) \) means substituting \( 4 \) for \( x \), resulting in the calculation: \[ p(4) = \frac{1}{2}(4)^4 - 2(4)^2 + 4 \]
• Similarly, evaluating \( p(-2) \) involves replacing \( x \) with \( -2 \), leading to: \[ p(-2) = \frac{1}{2}(-2)^4 - 2(-2)^2 + 4 \]

Function evaluation helps us understand what the output will be for certain inputs which is foundational in analyzing and graphing functions.

Substitution Method

The substitution method is a fundamental technique used in algebra to evaluate expressions, equations, or functions by replacing variables with specific values. This method allows us to simplify complex algebraic expressions.
When working with polynomial functions, the substitution method is essential. Here's a breakdown of how to apply it:

• Identify the variable in the function or expression. In our example, the variable is \( x \).
• Choose the values you want to substitute. For the problem, these were \( 4 \) and \( -2 \).
• Replace every instance of the variable with the chosen value, ensuring all operations follow the order of operations (PEMDAS/BODMAS rules).

By substituting values systematically, you can solve each part of the polynomial separately, making the problem easier to handle.

Polynomial Functions

Polynomial functions are expressions involving variables and coefficients, structured with one or more terms. Each term consists of a coefficient multiplied by a variable raised to a non-negative integer exponent.
For example, the function in the exercise, \[ p(x) = \frac{1}{2}x^4 - 2x^2 + 4 \] is a polynomial function with three terms:

• The first term is \( \frac{1}{2}x^4 \), which is a fourth-degree term.
• The second term is \( -2x^2 \), a second-degree term.
• The constant term is \( 4 \).

Polynomial functions can model various real-world situations and are central to many branches of mathematics. They can have multiple variables or just one, like in this case, and understanding their structure and how to manipulate them is key in solving algebraic problems.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Unlike a polynomial function, an algebraic expression doesn't always imply a function or equation, but it forms the building blocks for creating them.
In our specific exercise, the expression involves:

• Arithmetic operations like addition, subtraction, and multiplication.
• Exponents, where the variable \( x \) is raised to a power.

Managing algebraic expressions requires familiarity with basic algebra rules, like combining like terms and distributing coefficients. By manipulating these expressions efficiently, we can solve equations and understand relationships within functions.
Ultimately, grasping algebraic expressions helps in evaluating functions, especially when identifying components such as the constant and coefficients in polynomial terms. This proficiency is critical to simplify expressions and perform accurate calculations in algebra.