Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. The graph of a quadratic model with a positive leading coefficient will have a minimum value at its vertex.

Step-by-Step Solution

Answer

The statement is true. The graph of a quadratic model with a positive leading coefficient will indeed have a minimum value at its vertex, because the parabola opens upwards and the vertex of the parabola represents the lowest point of the graph.

Step 1: Understanding the Quadratic Model

The standard form of a quadratic equation is \(ax^2 + bx + c\). Here, \(a\) is the leading coefficient, \(b\) is the coefficient of \(x\), and \(c\) is the constant.

Step 2: Analyze the Effect of Leading Coefficient

The sign of the leading coefficient \(a\) in a quadratic function determines the direction of the parabola that represents the function graphically. Specifically, a positive leading coefficient gives a parabola which opens upwards and a negative leading coefficient gives a parabola which opens downwards.

Step 3: Determine the Location of Vertex

The vertex of a parabola concerning the quadratic function \(f(x) = ax^2 + bx + c\) occurs at the point \((-b/(2a), f(-b/(2a)))\). If the parabola opens upwards, the vertex represents the minimum point on the graph, because the values of \(f(x)\) for \(x\) not equal to \(-b/(2a)\) are greater than the value at the vertex. Conversely, if the parabola opens downwards, the corresponding vertex represents the maximum value of the function.

Step 4: Validate the Statement

Considering Steps 2 and 3, if the quadratic function has a positive leading coefficient, the parabola opens upwards and the vertex represents the minimum value of the function.

Key Concepts

Leading Coefficient

In a quadratic function, the leading coefficient is quite significant. It is the number in front of the variable raised to the second power, commonly represented by 'a' in the standard quadratic expression: \(ax^2 + bx + c\). The value of this coefficient influences the shape and direction of the parabola formed by the quadratic function.

• If the leading coefficient is positive \((a > 0)\), the parabola will open upwards, resembling a U-shape.
• Conversely, if the leading coefficient is negative \((a < 0)\), the parabola opens downwards, forming an inverted U-shape.

Being aware of this helps in predicting and analyzing the graph of any given quadratic function by simply inspecting the sign of this coefficient.

Vertex of a Parabola

The vertex of a parabola is a pivotal concept when discussing quadratic functions. It is the point at which the parabola changes direction. Specifically, for the quadratic function represented by \(f(x) = ax^2 + bx + c\), the vertex is located at the coordinates \((-b/(2a), f(-b/(2a)))\).

• When the parabola opens upwards \((a > 0)\), this vertex is the lowest point on the graph, often described as the minimum value of the function.
• On the other hand, if the parabola opens downward \((a < 0)\), the vertex is the highest point on the graph, the maximum value of the function.

By calculating the vertex, we can easily determine critical information about the behavior and characteristics of the function.

Graph of a Parabola

Graphing a parabola gives a visual representation of the quadratic function. The structure of the graph is largely dictated by the leading coefficient and the vertex. Here are a few key guiding points:

• The direction in which the parabola opens is dictated by the leading coefficient. Positive opens upwards, forming a U-shape, while negative opens downwards, forming an inverted U.
• The vertex represents either the minimum or maximum point, providing insight into where the function achieves these extrema.
• The symmetry of the parabola is around its vertical line through the vertex, known as the axis of symmetry, given by \(x = -b/(2a)\).

Understanding these elements helps in plotting the graph accurately and predicting how changes in the equation affect its overall appearance.