Problem 65 - Algebra 1: Concepts and Skills

Question

Evaluate the function when x 0, 1, 2, 3, and 4. \(f(x)=\frac{x^{2}}{2}\)

Step-by-Step Solution

Answer

The function \(f(x) = \frac{x^2}{2}\) evaluates to 0, 0.5, 2, 4.5, 8 when \(x = 0, 1, 2, 3, 4\) respectively.

Step 1: Substitute x=0

Substitute \(x = 0\) into the equation: \(f(0) =\frac{0^{2}}{2}= 0\)

Step 2: Substitute x=1

Substitute \(x = 1\) into the equation: \(f(1) =\frac{1^{2}}{2}= 0.5\)

Step 3: Substitute x=2

Substitute \(x = 2\) into the equation: \(f(2) =\frac{2^{2}}{2}= 2\)

Step 4: Substitute x=3

Substitute \(x = 3\) into the equation: \(f(3) =\frac{3^{2}}{2}= 4.5\)

Step 5: Substitute x=4

Substitute \(x = 4\) into the equation: \(f(4) =\frac{4^{2}}{2}= 8\

Key Concepts

Quadratic Function

A quadratic function is a special type of mathematical expression, represented typically as \( f(x) = ax^2 + bx + c \). The key characteristic of a quadratic function is that the highest exponent of the variable is 2. This quadratic nature makes the graph of the function a parabola, which can either open upwards or downwards. In the case of our exercise, the quadratic function is simplified to \( f(x) = \frac{x^2}{2} \), where the term \( ax^2 \) is represented by \( \frac{x^2}{2} \), and terms \( bx \) and \( c \) are absent. This simplifies the evaluation process greatly. Understanding this form helps comprehend why evaluating at different \( x \) values results in smooth, predictable changes.

Substitution Method

The substitution method involves replacing a variable with a specific value to evaluate a function. It's a straightforward process that's crucial for solving algebraic equations and evaluating functions like quadratics. In our example, we are substituting different values of \( x \) into the function \( f(x) = \frac{x^2}{2} \).

• First, you identify the variable in the equation to be substituted, which is \( x \).
• Next, replace \( x \) with each value given (0, 1, 2, 3, and 4).
• Calculate the result for each substitution to find the function's output.

This method is not only useful in quadratic functions but also in other types of equations where direct calculation is possible once substitution is applied.

Mathematical Expressions

Mathematical expressions consist of numbers, variables, and operations that collectively form part of an equation or function. In essence, these are the language of algebra that mathematicians use to convey quantities and relationships. In our exercise, \( \frac{x^2}{2} \) is a mathematical expression that defines our function.Understanding components like:

• Numerators and denominators in fractions
• Exponents, indicating the power to which a number is raised
• Variables, which are letters representing unknown values

are vital for interpreting and working with expressions. Mastering these concepts allows you to manipulate expressions to solve for unknowns or to transform them in various algebraic processes.

Algebraic Equations

Algebraic equations are mathematical statements that use an equal sign to show the equality between two expressions. They are fundamental tools in algebra for solving problems and representing real-world situations mathematically. In the function we are evaluating \( f(x) = \frac{x^2}{2} \), though not an equation in the traditional sense (as it doesn't solve for zero or another outcome), understanding its form is crucial.Key points about algebraic equations:

• Equations balance perfectly, meaning whatever operation you do to one side must be mirrored on the other.
• In quadratic equations, finding roots involves setting the quadratic equal to zero. Though our task didn't involve solving an equation, interpreting function values teaches foundational skills useful for all algebraic processes.

Learning to manage these equations helps predict outcomes and solve for unknowns efficiently.