Problem 22 - Algebra 2

Question

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=3 x^{2} $$

Step-by-Step Solution

Answer

The function has a minimum value of 0 at \( x = 0 \). Domain: all real numbers. Range: \( y \geq 0 \).

Step 1: Identify the function type

The given function is a quadratic function of the form \( ax^2 \), where \( a = 3 \). A quadratic function with a positive coefficient \( a \) opens upwards and has a minimum value.

Step 2: Determine the vertex

For the function \( f(x) = 3x^2 \), since it is centered at the origin and in the form \( ax^2 \), the vertex is at \( (0, 0) \). This vertex represents the point of minimum value.

Step 3: Identify the minimum value

The minimum value of the function occurs at the vertex, so the minimum value of \( f(x) = 3x^2 \) is \( 0 \).

Step 4: State the domain of the function

The domain of the quadratic function \( f(x) = 3x^2 \) is all real numbers \( \mathbb{R} \) because you can substitute any real number for \( x \) in the function.

Step 5: Determine the range of the function

Since the function has a minimum value at \( y = 0 \) and opens upwards, the range is all real numbers \( y \geq 0 \).

Key Concepts

Vertex

In a quadratic function like \( f(x) = 3x^2 \), identifying the vertex is the key to understanding its graph. A vertex is the point where the function reaches either a peak or a trough. It acts like the pivot around which the parabola is shaped. For a function in the form \( ax^2 \), the vertex is located at the origin, or \( (0, 0) \), because no horizontal or vertical shifts are present. The vertex in a quadratic function reflects the core value where the change in direction occurs, making it crucial for understanding the nature of the curve. In this instance, since \( a \) is positive, the vertex is the lowest point on the graph, forming a trough.

Minimum Value

The minimum value of a quadratic function is simply the smallest output that the function can produce. In the case of \( f(x) = 3x^2 \), because the parabola opens upwards due to the positive coefficient \( a = 3 \), it means the graph will have a trough or low point. This low point occurs at the vertex. In this function, the minimum value is \( 0 \) at the vertex \( (0, 0) \).

• The parabola opens upwards as \( a \) is a positive value.
• The minimum value is always situated at the coordinate of the vertex for these types of quadratic functions.

Understanding this concept helps provide insight into many real-life situations modeled by parabolas, such as projectile motion and optimization problems.

Domain and Range

The domain and range of a function describe which values \( x \) and \( f(x) \) can take on, respectively. For any quadratic function like \( f(x) = 3x^2 \), you can substitute any real number for \( x \), and thus its domain is all real numbers, represented by \( \mathbb{R} \).

• Domain: This is the set of all possible input values for the function and for \( f(x) = 3x^2 \), the domain is \(-\infty < x < \infty\).
• Range: This describes all possible outputs. Since this quadratic has a minimum value at \( f(x) = 0 \) and opens upwards, the range is \( y \ge 0 \).

These concepts help us fully understand the behavior and limitations of functions, allowing better predictions and interpretations of real-world scenarios modelled by quadratic equations.