Graphs of Quadratic Functions

Understanding Quadratic Functions

A quadratic function is a type of polynomial that can be written in the form:

$$f(x) = ax^2 + bx + c$$

where:

• a, b, and c are constants
• a ≠ 0 (if a = 0, the function becomes linear)

The graph of a quadratic function is a parabola.

Key Features of Quadratic Graphs

• Vertex: The highest or lowest point on the graph, depending on the direction of the parabola.
• Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
• Direction: If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
• Roots: The points where the graph intersects the x-axis, also known as x-intercepts or solutions of the equation $ax^2 + bx + c = 0$.

Steps to Plot a Quadratic Graph

1. Create a Table of Values: Choose a range of x-values and calculate the corresponding y-values using the quadratic equation.
2. Plot the Points: On a coordinate grid, plot the points from your table.
3. Draw the Parabola: Connect the points with a smooth curve to form the parabola.
4. Identify Key Features: Mark the vertex, axis of symmetry, and roots.

Example

Example: Plot the graph of $y = x^2 - 4x + 3$