Problem 6 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=-4 x^{2} $$

Step-by-Step Solution

Answer

The vertex of the parabola \( f(x) = -4x^2 \) is at (0, 0).

Step 1: Understand the Standard Form of a Parabola

The standard form of a parabola is given by the equation: \[ f(x) = ax^2 + bx + c \] Here, the given equation is \( f(x) = -4x^2 \). Notice there are no \( bx \) or \( c \) terms, meaning \( b = 0 \) and \( c = 0 \).

Step 2: Identify the Coefficients

From the given function \( f(x) = -4x^2 \), identify the coefficients: \[ a = -4, \, b = 0, \, c = 0 \]

Step 3: Calculate the Vertex x-value

The x-coordinate of the vertex for a parabola in standard form is given by: \[ x = -\frac{b}{2a} \]Substitute the values of \( b \) and \( a \) into the equation: \[ x = -\frac{0}{2(-4)} = 0 \]

Step 4: Calculate the Vertex y-value

Substitute the x-value of the vertex back into the function to find the y-coordinate. With \( x = 0 \), \[ f(0) = -4(0)^2 = 0 \]So, the y-coordinate of the vertex is 0.

Step 5: Write the Vertex Coordinates

Combine the x and y values to write the vertex in coordinate form: \[ (0, 0) \]

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function where the highest degree of the variable is squared. These functions are usually written in the form \[ f(x) = ax^2 + bx + c \]. In this formula:

• \