Problem 8 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=\frac{1}{2} x^{2} $$

Step-by-Step Solution

Answer

The vertex is (0,0).

Step 1: Identify the General Form

The general form of a quadratic function is given by \[f(x) = ax^2 + bx + c\] where \( a \), \( b \), and \( c \) are constants.

Step 2: Compare with the Given Function

Compare the given function \( f(x) = \frac{1}{2} x^2 \) with the general form. Here, \( a = \frac{1}{2} \), \( b = 0 \), and \( c = 0 \).

Step 3: Calculate the Vertex Coordinates

The vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \] where \( (h, k) \) is the vertex. For the vertex calculation in standard form, the formula for the vertex \( h \) (or x-coordinate) is \( h = -\frac{b}{2a} \). Given \( b = 0 \) and \( a = \frac{1}{2} \), we find: \[ h = -\frac{0}{2 \times \frac{1}{2}} = 0 \]. Then, substitute \( 0 \) for \( x \) in the function to find \( k \): \[ k = f(0) = \frac{1}{2} (0)^2 = 0 \]. Thus, the vertex is \( (0, 0) \).

Key Concepts

Quadratic Functions

Quadratic functions describe the relationship between the variables in the form of a parabola. Every quadratic function can be written in the form \( f(x) = ax^2 + bx + c \). Here:

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