Problem 5 - Beginning and Intermediate Algebra

Question

Identify the vertex of each parabola. $$ f(x)=-3 x^{2} $$

Step-by-Step Solution

Answer

The vertex is \( (0, 0) \).

Step 1: Understand the Standard Form

Parabolas can be written in the standard form: \[ f(x) = ax^2 + bx + c \].The vertex form of a quadratic function is \[ f(x) = a(x-h)^2 + k \],where \((h, k)\) is the vertex.

Step 2: Identify Coefficients

For the given function \[ f(x) = -3x^2 \],the coefficients are: \(a = -3\), \(b = 0\), and \(c = 0\).

Step 3: Calculate Vertex (h, k)

The formula to find the vertex \((h, k)\) is \[ h = -\frac{b}{2a} \]and \[ k = f(h) \].Here, \( b = 0 \), so\[ h = -\frac{0}{2(-3)} = 0 \].Substitute \( h = 0 \)back into the function to find \( k \):\[ k = f(0) = -3(0)^2 = 0 \].So the vertex is \((0, 0)\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial that is characterized by its highest degree being two. This means the variable (commonly represented as x) is squared. Quadratic functions are commonly written in the form f(x) = ax^2 + bx + c Here,

• a is the coefficient of the squared term,
• b is the coefficient of the linear term,
• c is the constant term.

Quadratic functions graph as parabolas, which are U-shaped curves. These can open upwards or downwards depending on the sign of the coefficient a. If a > 0, the parabola opens upwards. If a < 0, it opens downwards. Understanding the structure of quadratic functions is key to analyzing their properties, such as finding the vertex.

Standard Form

The standard form of a quadratic function is: f(x) = ax^2 + bx + c This form is useful because it provides a straightforward way to identify the coefficients a, b, and c. For example, in the function given in the exercise, f(x) = -3x^2 we can see that:

• a = -3
• b = 0
• c = 0

Identifying these coefficients is an essential step before moving to vertex calculation. The standard form clearly presents the basic elements needed for further analysis, such as determining the direction the parabola opens and calculating the vertex.

Vertex Form

Another helpful way to write a quadratic function is in vertex form: f(x) = a(x-h)^2 + k In this form,

• (h, k) represents the vertex of the parabola.
• a still determines the direction and width of the parabola.

The vertex form is particularly useful because it allows you to easily identify the vertex without any further calculations. Converting from standard form to vertex form may involve completing the square, but once you have the vertex form, the vertex of the parabola is directly visible. In our exercise, converting f(x) = -3x^2 to vertex form would show that the vertex is at (0, 0), since the equation fits the structure of a(x-h)^2 + k with h=0 and k=0.

Vertex Calculation

To find the vertex of a quadratic function in standard form, you can use a simple formula. For the standard form f(x) = ax^2 + bx + c the vertex (h, k) can be found using:

• h = -b/(2a)
• k = f(h)

For the given function f(x) = -3x^2 we have a = -3, b = 0, and c = 0. Using the formula:

• h = -0/(2*-3) = 0
• k = f(0) = -3(0)^2 = 0

Therefore, the vertex is at (0, 0). Calculating the vertex is an essential skill, as it allows you to identify the highest or lowest point of the parabola. This is critical in graphing the parabola and understanding its behavior.