Axis of Symmetry for Parabolas: A Comprehensive Guide

A parabola is a conic section that is symmetrical about a particular line. This line is known as the axis of symmetry. In this article, we will explore the concept of the axis of symmetry and how it relates to the x-intercepts of a parabola. We will use specific examples to illustrate the process of finding the axis of symmetry for a parabola with x-intercepts at -3 and 7.

Understanding the Axis of Symmetry

The axis of symmetry for a parabola is the line that divides the parabola into two reflective halves. This line is perpendicular to the x-axis and passes through the vertex of the parabola. For a parabola with x-intercepts, the axis of symmetry is the perpendicular bisector of the segment joining these intercepts.

Example Problem

Consider a parabola with x-intercepts at -3 and 7. To find the axis of symmetry, we need to calculate the midpoint of these intercepts.

Finding the Midpoint

The formula for the midpoint of two points (x_1, y_1) and (x_2, y_2) on the x-axis is given by:

Midpoint left( frac{{x_1 x_2}}{2}, 0 right)

Applying this formula to our x-intercepts:

Midpoint left( frac{{-3 7}}{2}, 0 right) left( frac{4}{2}, 0 right) (2, 0)

Therefore, the axis of symmetry is the vertical line passing through the point (2, 0), represented by the equation:

x 2

Mathematical Derivation

Let's consider a parabola given by the equation:

y ax^2 bx c

If the x-intercepts are at -3 and 7, the roots of the equation are:

x -3 and x 7

The axis of symmetry of a parabola is the average of its roots. Therefore, the x-coordinate of the axis of symmetry is given by:

x_{sym} frac{{x_1 x_2}}{2}

Substituting the intercepts:

x_{sym} frac{{-3 7}}{2} frac{4}{2} 2

Hence, the axis of symmetry is:

x 2

Proving the Axis of Symmetry

To further understand the concept, let's consider the equation of a parabola in its standard form:

y a(x - h)^2 k

Where (h, k) is the vertex of the parabola. The axis of symmetry is the vertical line passing through the vertex, which is represented by:

x h

Now, let's rewrite the general form of a quadratic equation:

y ax^2 bx c

Using the vertex form, we can find the vertex by completing the square:

a(x^2 frac{b}{a}x) c 0

a[(x frac{b}{2a})^2 - (frac{b}{2a})^2] c 0

a(x frac{b}{2a})^2 - a(frac{b}{2a})^2 c 0

a(x frac{b}{2a})^2 - frac{b^2}{4a} c 0

Therefore, the axis of symmetry is:

x -frac{b}{2a}

For the given example with x-intercepts at -3 and 7, the axis of symmetry is:

x -frac{{-3 7}}{2a} frac{4}{2a} 2

Conclusion

The axis of symmetry for a parabola with x-intercepts at -3 and 7 is x 2. This line divides the parabola into two symmetric halves. The concept of the axis of symmetry is crucial in understanding the properties and behavior of parabolas. By knowing the axis of symmetry, we can more easily analyze and graph parabolas and solve related problems.