Problem 76 - Algebra 2

Question

Find each function value. \(h(x)=2 x^{2}-4, h(0)\)

Step-by-Step Solution

Answer

The value of the function \(h(0)\) is \(-4\).

Step 1: Understand the Problem

We need to find the value of the function \(h(x) = 2x^2 - 4\) when \(x\) is equal to 0.

Step 2: Substitute in the Function

Substitute \(x = 0\) into the function. The function is \(h(x) = 2x^2 - 4\). So, replace \(x\) with 0: \(h(0) = 2(0)^2 - 4\).

Step 3: Simplify the Expression

Calculate \((0)^2\), which is 0, and substitute it back into the equation: \(h(0) = 2 \times 0 - 4\).

Step 4: Solve the Expression

Multiply 2 by 0, which gives 0: \(h(0) = 0 - 4\). Continue by simplifying the subtraction: \(h(0) = -4\).

Step 5: Conclusion

Thus, the value of the function \(h(0)\) is \(-4\).

Key Concepts

Substitution

In mathematics, substitution is a fundamental concept often used to calculate values, solve equations, or evaluate expressions by replacing variables with numbers or other expressions. When we talk about function evaluation, substitution plays a crucial role.
To evaluate the function \(h(x) = 2x^2 - 4\) at \(x = 0\), we substitute \(0\) for \(x\) in the expression. This means every place where \(x\) appears in the equation, it is replaced with \(0\).
For instance:

• Start with the function: \(h(x) = 2x^2 - 4\).
• Substitute \(x = 0\): \(h(0) = 2(0)^2 - 4\).

Substitution allows us to assess the value of the function for a particular input, streamlining the calculation process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators such as addition, subtraction, multiplication, and division. These are the building blocks of algebra and are used to represent relationships and calculations concisely.
In the given exercise, the function \(h(x) = 2x^2 - 4\) is an algebraic expression. It includes:

• Terms: \(2x^2\) and \(-4\)
• Coefficient: the number \(2\) which is multiplied by the variable part \(x^2\)
• Constant: \(-4\), a term that remains unchanged since it lacks a variable part

Understanding the structure of algebraic expressions helps simplify and evaluate them efficiently, especially while performing substitution.

Quadratic Functions

Quadratic functions are polynomial functions characterized by the highest power of the variable being two. A general form of a quadratic function is \(ax^2 + bx + c\).
In our exercise, the function \(h(x) = 2x^2 - 4\) is a classic example of a quadratic function. This specific function is expressed with:

• A leading coefficient \(a\) of \(2\), attached to \(x^2\)
• Zero for the linear term \(b\)
• A constant term \(c\) of \(-4\)

Quadratic functions graph as parabolas, which can open upwards or downwards. The coefficients determine the shape and position of the graph. Evaluating these at specific points, like in this exercise, helps us understand their behavior.