Introduction

Graphing quadratic functions is an essential skill in understanding the behavior of parabolic curves. These functions are defined by an equation of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Steps to Graphing Quadratic Functions

1. Identify the vertex: The vertex, denoted by (h, k), is the turning point of the parabola. The h-coordinate of the vertex can be found using the formula h = -b / (2a), and the k-coordinate can be obtained by substituting the h-value into the equation.

2. Find the axis of symmetry: The axis of symmetry is a vertical line that passes through the vertex. Its equation is x = h.

3. Determine the y-intercept: Substitute x = 0 into the equation to find the value of f(0).

4. Plot additional points: Choose a few x-values on either side of the axis of symmetry, substitute them into the equation, and calculate the corresponding y-values. These points will help create a smooth curve.

5. Sketch the parabola: Connect the plotted points with a smooth curve that opens upward (a > 0) or downward (a < 0).

Example

Let's consider the quadratic function f(x) = x^2 - 4x + 3. By applying the steps outlined above:

1. The vertex is located at (h, k) = (2, -1).
2. The axis of symmetry is x = 2.
3. The y-intercept is f(0) = 3.
4. Choosing x = 1 and x = 3, we find the corresponding y-values to be f(1) = 0 and f(3) = 0 respectively.

By plotting these points and connecting them, we obtain the graph of f(x) = x^2 - 4x + 3.

Remember, practice makes perfect! By graphing quadratic functions regularly, you will gain confidence in understanding their behavior.