Problem 100 - Brief Applied Calculus

Question

Explain why, if a quadratic function has two \(x\) intercepts, the \(x\) -coordinate of the vertex will be halfway between them.

Step-by-Step Solution

Answer

The x-coordinate of the vertex is the midpoint between the x-intercepts, calculated as \(-\frac{b}{2a}\).

Step 1: Understand the Structure of a Quadratic Function

A quadratic function is generally expressed as \(f(x) = ax^2 + bx + c\). This function forms a parabola when plotted. The function can have up to two \(x\)-intercepts, which are the solutions to the equation \(ax^2 + bx + c = 0\). These are also known as the roots of the quadratic function.

Step 2: Identify the Quadratic Formula

The solutions for the \(x\)-intercepts (roots) of the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula: \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\). These solutions give the \(x\)-coordinates where the quadratic function intersects the x-axis.

Step 3: Define the X-intercepts

The two \(x\)-intercepts can be labeled as \(x_1\) and \(x_2\). Using the quadratic formula, these intercepts are \(x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\) and \(x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\).

Step 4: Calculate the Vertex X-coordinate

The \(x\)-coordinate of the vertex of a quadratic function is given by the formula \(x = -\frac{b}{2a}\). This formula provides the x-value that is exactly halfway between the two x-intercepts.

Step 5: Establish Relationship Between Vertex and X-intercepts

Consider the midpoint formula for the \(x\)-intercepts \(x_1\) and \(x_2\), which is \(\frac{x_1 + x_2}{2}\). Substituting for \(x_1\) and \(x_2\), we get:\[\frac{\frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}}{2} = \frac{-b}{2a}\]This simplifies to \(-\frac{b}{2a}\), the \(x\)-coordinate of the vertex. Hence, the vertex is exactly halfway between the \(x\)-intercepts.

Key Concepts

x-intercepts

The x-intercepts of a quadratic function are the points where the graph of the parabola crosses the x-axis. These points are significant because they represent the values of x that make the quadratic function equal to zero. Thus, finding the x-intercepts involves solving the equation of the quadratic function set to zero:
\( ax^2 + bx + c = 0 \).
To solve this equation, we often use the quadratic formula, which provides us with the solutions or roots of the equation:

• \( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \)
• \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \)

These solutions identify the two x-intercepts on the parabola. Remember, the x-intercepts demonstrate where the parabola touches or crosses the x-axis. Understanding these points is essential for analyzing the shape and position of a parabola on the graph.

vertex of a parabola

The vertex of a parabola is the point where the curve changes direction—this is known as the 'turning point'. In a quadratic function's graph, this vertex can be either a maximum or a minimum point, depending on the parabola's orientation.
The x-coordinate of the vertex can be quickly calculated using the formula:
\( x = -\frac{b}{2a} \).
This is significant because it tells us where the exact middle between the two x-intercepts is located.
The y-coordinate of the vertex can be found by substituting the x value back into the original quadratic equation. Once both coordinates are known, the vertex gives us crucial information about the parabola:

• It identifies the line of symmetry for the parabola.
• It represents the optimal lowest or highest point depending on the parabola's direction.

This makes the vertex an integral part of understanding and graphing quadratic functions.

quadratic formula

The quadratic formula is a powerful tool that allows us to find the x-intercepts of a quadratic equation easily. The formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]provides solutions to the quadratic equation \(ax^2 + bx + c = 0\). Here's a breakdown of each component:

• -b: The opposite of the linear coefficient.
• ± \(\sqrt{b^2 - 4ac}\): The discriminant, which indicates the nature of the roots (real and distinct, real and repeated, or complex).
• 2a: The denominator which normalizes the value of x based on the leading coefficient.

Being able to use the quadratic formula effectively unlocks the ability to solve any quadratic equations you encounter. It's particularly useful when the quadratics are not easily factorable. By understanding each part of the formula, you can also predict the number and type of solutions you might expect for the equation.

parabola

A parabola is the graphical representation of a quadratic function, creating a U-shaped curve. Each parabola appears based on the general form of the quadratic equation \( f(x) = ax^2 + bx + c \).
Depending on whether the coefficient \( a \) is positive or negative, the parabola will open upwards or downwards, respectively.

• If \( a > 0 \): The parabola opens upwards, and the vertex represents the minimum point.
• If \( a < 0 \): The parabola opens downwards, and the vertex is the maximum point.

The parabola's shape and position are defined by the coefficients \( a \), \( b \), and \( c \), influencing the width, tilt, and vertical position of the graph. Key characteristics of parabolas include:

• Axis of symmetry: A vertical line passing through the vertex that divides the parabola into two symmetrical halves.
• Direction: Determined by the sign of \( a \).
• Vertex: The peak or the lowest point, crucial for defining the parabola's transformational attributes.

Understanding these aspects allows you to effectively graph and interpret every quadratic function through the lens of its parabolic shape.