Question

WRITING A quadratic function is increasing to the left of x = 2 and decreasing to the right of x = 2. Will the vertex be the highest or lowest point on the graph of the parabola? Explain.

Short answer

The vertex will be the highest point on the graph of the parabola.

Step-by-step solution

Understanding a Quadratic Function's Behavior

A quadratic function would take the general form \(y = ax^2 + bx + c\). When the leading coefficient 'a' is positive, the parabola opens upwards which means it has a minimum point. If 'a' is negative, the parabola opens downwards which means it has a maximum point.

Interpreting the Problem Statement

The question states that the function is increasing to the left of x = 2 and decreasing to the right of x = 2. This describes a parabola that opens downwards because it is going from high values to low values as x increases.

Infer the Nature of the Vertex

Based on the above analysis, it can be concluded that since the parabola is opening downwards the vertex will be the highest point on the graph.

Key concepts

Parabolas

Parabolas are U-shaped graphs that represent quadratic functions, characterized by an equation of the form \(y = ax^2 + bx + c\). Quadratic functions are incredibly important in algebra, as they frequently pop up in both mathematics problems and real-world applications. The shape and direction of a parabola are determined by the coefficient \(a\). A few key traits define the nature of a parabola:

• If \(a > 0\), the parabola opens upwards, creating a U-shaped curve. The lowest point in this case is called the vertex.
• If \(a < 0\), the parabola opens downwards, resulting in an upside-down U shape. Here, the vertex is the highest point.

Understanding how the value of \(a\) influences the direction of the parabola is crucial to interpreting and predicting the behavior of quadratic functions. Parabolas are symmetric, and the line of symmetry passes vertically through the vertex.

Vertex of a Parabola

The vertex of a parabola is a critical point that represents either the maximum or minimum value of the quadratic function, depending on the direction the parabola opens. This point lies exactly in the middle of the parabola's curve and on its axis of symmetry.

The vertex can be calculated using the formula \(x = -\frac{b}{2a}\), where \(b\) and \(a\) are coefficients from the quadratic equation \(y = ax^2 + bx + c\). Once you have the \(x\)-coordinate, you can substitute it back into the equation to find the \(y\)-coordinate.

Key points to remember about the vertex include:

• For an upward-opening parabola, the vertex is the minimum point.
• For a downward-opening parabola, the vertex is the maximum point.
• The vertex serves as an indicator of the parabola's highest or lowest value and helps in graphing the function accurately.

Graphing Quadratic Equations

Graphing quadratic equations involves visually representing the behavior of quadratic functions on a coordinate grid. The graph of a quadratic equation is a parabola, and several key steps can help in plotting these graphs.

First, identify whether the parabola opens upwards or downwards by checking the sign of the coefficient \(a\). Knowing the direction is crucial for accurately depicting the vertex as a high or low point.

Next, calculate the vertex using the formula \(x = -\frac{b}{2a}\). Then, substitute into the quadratic equation to get the \(y\) value. With the vertex identified, draw the axis of symmetry—a vertical line through the vertex.

Plotting additional points on either side of the vertex will help define the shape of the parabola. These points should be symmetrically placed concerning the axis of symmetry.

Important considerations while graphing:

• The vertex and the direction of the parabola provide the framework for the graph.
• The graph should mirror over the axis of symmetry, stating the symmetry of quadratic functions.
• Choose test points to accurately reflect the parabola's curvature.

Graphing not only aids in visualizing the function but also enhances understanding of properties like intercepts, vertex positions, and the general trend of the parabola.