Problem 86 - Algebra 1

Question

Writing Explain how you can use this two-part form of the quadratic formula $$x=\frac{-b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a}$$ to find the distance between the axis of symmetry of a parabola and either of its \(x\) -intercepts.

Step-by-Step Solution

Answer

The distance between the axis of symmetry of a parabola and its x-intercepts is the value defined by the second part of the quadratic formula \(\frac{\sqrt{b^2 - 4ac}}{2a}\). This is the amount added to or subtracted from the x-coordinate of the vertex (which is \(\frac{-b}{2a}\) and represents the axis of symmetry) to obtain the two different x-intercepts of the parabola.

Step 1: Understand the structure of a Parabola

A parabola, graphed from a quadratic equation, generally opens either upwards or downwards and has a single line of symmetry known as the axis of symmetry (AoS). The AoS intersects the parabola at its vertex, and also lies equidistant from its two roots (if they exist), also known as x-intercepts.

Step 2: Express the Quadratic Formula in two parts

The quadratic formula for any quadratic equation given as \(ax^2 + bx + c = 0\) is expressed as \(x = \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}\).This formula obtains its form from the method of completing the square and is compound of two different parts: the first is the \(\frac{-b}{2a}\) term, which gives the x-coordinate of the vertex (and AoS) of the parabola, and the second part \(\pm \frac{\sqrt{b^2 - 4ac}}{2a}\) which provides the distance from the vertex to either root. Hence, to find the roots, we combine both parts.

Step 3: Find the distance from AoS to x-intercepts

The distance from the axis of symmetry to any of the x-intercepts will be the value defined by the second part of the quadratic formula \(\frac{\sqrt{b^2 - 4ac}}{2a}\). This can be derived by imagining the position of the terms in the quadratic formula with reference to the graph of a parabola. Essentially, the \(\pm\) in the second part of the equation allows us to get the two different x-intercepts of the parabola by adding or subtracting from the AoS x-coordinate, thus giving the distance from AoS to each root.

Key Concepts

Understanding the Axis of Symmetry

In the world of parabolas, the axis of symmetry (AoS) plays a crucial role. It is the invisible line that perfectly splits the parabola into two symmetrical halves. To identify the axis of symmetry in a quadratic equation, you can use the formula:

• \( x = \frac{-b}{2a} \)

This equation comes from the first part of the quadratic formula. It helps you find the x-coordinate of the parabola's vertex, which is the highest or lowest point of the parabola depending on its orientation. Keep in mind that whether your parabola opens upwards (like a smile) or downwards (like a frown), the AoS will always pass through the vertex.
In simpler terms, the axis of symmetry is like a vertical mirror line for your parabola, ensuring each side is a mirror image of the other.

Decoding the X-intercepts

When a parabola crosses the x-axis, these points of intersection are known as x-intercepts, or roots. Finding them is essential because they provide solutions to the quadratic equation represented by the parabola.
To uncover these x-intercepts, we look to the quadratic formula once again:

• \( x = \frac{-b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)

The two solutions are indicated by the "\( \pm \)" sign in the formula. This sign tells us that there are potentially two different roots: one is found by adding the square root portion, and the other by subtracting it.
The x-intercepts are not just important mathematically; they also determine where the parabola "touches" the x-axis on a graph.

Measuring the Distance from Axis of Symmetry to X-intercepts

The gap between the axis of symmetry and the x-intercepts is just as significant as the terms themselves. In the formula, this distance is revealed by the second half of the expression:

• \( \frac{\sqrt{b^2 - 4ac}}{2a} \)

This formula part comes after the "\( \pm \)" and tells us how far the x-intercepts are from the AoS on both sides of the vertex. Essentially, it reflects the horizontal distance from the axis of symmetry to where the parabola crosses the x-axis.
Whether the parabola's roots are real or imaginary can affect this distance—real roots make the distance visible on a graph while imaginary roots mean the parabola doesn’t touch the x-axis, and the distance becomes a conceptual aid for understanding.
Such comprehension of the distance is vital for solving various problems related to the quadratic equations and their graphical representations.