Quadratic functions are mathematical functions that involve a variable raised to the power of two, resulting in a parabolic shape when graphed. Graphing quadratic functions involves understanding key elements such as the vertex, axis of symmetry, and the direction the parabola opens.

To determine the vertex of a quadratic function, we can use the formula x = -b / (2a). The x-coordinate of the vertex is the value obtained from this formula, while the y-coordinate can be found by substituting the x-value into the original function. The axis of symmetry is the vertical line passing through the vertex.

To find the roots or x-intercepts of a quadratic function, we set the function equal to zero and solve for x. This can be done by factoring, completing the square, or using the quadratic formula.

Graphing quadratic functions involves plotting points based on the calculated vertex and roots. We can also identify whether the parabola opens upward or downward by analyzing the coefficient of x² in the function. If the coefficient is positive, the parabola opens upward, and if it is negative, the parabola opens downward.

Let's consider an example: The quadratic function f(x) = x² - 4x + 3. Using the formula for the vertex, we find that the x-coordinate is 2, and substituting this value, we get y = -1. This coordinates gives us the vertex (2, -1). By factoring, we find the roots x = 1 and x = 3. Plotting these points and drawing the parabola through them, we can visualize the graph of the quadratic function.

Remember that practice is key when it comes to graphing quadratic functions. So keep practicing and mastering the techniques, and you'll be able to graph any quadratic function with ease!