Problem 52 - Algebra 2

Question

For a quadratic equation of the form \((x-p)(x-q)=0\) , show that the axis of symmetry of the related quadratic function is located halfway between the \(x\) -intercepts \(p\) and \(q .\)

Step-by-Step Solution

Answer

The axis of symmetry is at \(x = \frac{p+q}{2}\), midway between the x-intercepts.

Step 1: Identify the X-intercepts

The x-intercepts of the quadratic equation \((x-p)(x-q) = 0\) are given by the solutions of the equation when it equals zero, which are \(x = p\) and \(x = q\).

Step 2: Calculate the Midpoint

The axis of symmetry for the parabola is located halfway between the x-intercepts. To find this point, calculate the midpoint of \(p\) and \(q\). The midpoint formula is \(midpoint = \frac{p+q}{2}\).

Step 3: Conclude the Position of the Axis of Symmetry

Based on the calculation, the axis of symmetry of the quadratic function defined by \((x-p)(x-q) = 0\) is located at \(x = \frac{p+q}{2}\). This line is vertical and passes through the midpoint of the x-intercepts.

Key Concepts

Understanding the Axis of Symmetry

In the world of quadratic equations, the axis of symmetry is a central concept. For a quadratic function, this axis is a vertical line that runs directly through the most balanced part of a parabola, effectively splitting it into mirror images. This means that every point on one side of the parabola has an equivalent point on the other side.
The axis of symmetry is especially useful because it helps us find the vertex of the parabola, which is its highest or lowest point. Notably, for a quadratic function expressed in the form \[(x-p)(x-q)=0\]the axis of symmetry is located exactly between the two roots \(p\) and \(q\).

• This special line can be found using the formula \(x = \frac{p+q}{2}\).
• The symmetry of the parabola around this axis makes calculations and graph interpretations simpler.

Features of a Quadratic Function

A quadratic function is one of the most fundamental types of polynomial functions. It is usually written in three main forms:

• Standard form: \(ax^2 + bx + c = 0\)
• Factored form: \((x-p)(x-q) = 0\)
• Vertex form: \(a(x-h)^2 + k = 0\)

Each of these forms reveals different properties of the function and is useful for different purposes when solving equations or graphing.
In the context of the factored form, where the function is expressed as \((x-p)(x-q) = 0\), the roots \(p\) and \(q\) are easy to identify. These roots reflect the intersection points where the graph of the parabola crosses the x-axis.
Quadratic functions always graph as a U-shaped curve called a parabola, which can open either upward or downward, depending on the sign of the leading coefficient \(a\). This orientation affects the position of the vertex and the direction in which the parabola stretches.

Applying the Midpoint Formula

The midpoint formula is a straightforward calculation, often used in geometry and algebra to find the point that is exactly in the middle of two other points on a line segment.
To apply this formula, you don't need complex calculations. You simply take two values, add them together, and divide by two. In symbols, it's written as:\[\text{Midpoint} = \frac{p+q}{2}\]This formula is extremely handy, especially when identifying the axis of symmetry in quadratic equations.
By using the values of the x-intercepts \(p\) and \(q\), the midpoint formula gives us the x-coordinate of the axis of symmetry at

• \(x = \frac{p+q}{2}\)

This single value tells us precisely where the vertical line, or axis of symmetry, lies.