Quadratic functions are mathematical expressions of the form f of x equals a x squared plus b x plus c. One of their most important properties is symmetry. Every quadratic function has an axis of symmetry, which is a vertical line that divides the parabola into two mirror-image halves. In this example, we see the function f of x equals x squared minus 2x plus 1, which has its axis of symmetry at x equals 1.
Here we can see two points A and B on the parabola. Point A is at coordinates negative 1, 4, and point B is at coordinates 3, 4. Notice that both points have the same y-coordinate of 4. This is because they are equidistant from the axis of symmetry at x equals 1. Point A is 2 units to the left of the axis, and point B is 2 units to the right. This symmetric property holds for any pair of points on a quadratic function.
To find the axis of symmetry for any quadratic function, we use the formula x equals negative b divided by 2a. For our function f of x equals x squared minus 2x plus 1, we identify a equals 1 and b equals negative 2. Substituting these values into the formula: x equals negative negative 2 divided by 2 times 1, which simplifies to 2 divided by 2, giving us x equals 1. This confirms that our axis of symmetry is the vertical line x equals 1.