Lesson 33: Graphing Quadratic Functions (Parabolas, Vertex, Axis of Symmetry)
Quadratic functions are functions of the form f(x) = ax² + bx + c. Their graphs are called parabolas. Understanding the vertex, axis of symmetry, and direction of opening helps in graphing them.
Properties of a Parabola
- Vertex: The highest or lowest point of the parabola. Coordinates: (h, k)
- Axis of Symmetry: Vertical line passing through the vertex, x = h
- Direction: If a > 0, opens upwards; if a < 0, opens downwards
- Y-intercept: At (0, c)
Finding the Vertex
For f(x) = ax² + bx + c, the vertex formula is:
h = −b / (2a)
k = f(h)
Example: f(x) = 2x² − 4x + 1
- h = −(−4)/(2*2) = 4/4 = 1
- k = f(1) = 2(1)² − 4(1) + 1 = 2 − 4 + 1 = −1
- Vertex: (1, −1)
- Axis of symmetry: x = 1
- Since a = 2 > 0, parabola opens upward
Steps to Graph a Quadratic Function
- Find the vertex using h = −b/2a and k = f(h)
- Identify axis of symmetry: x = h
- Determine direction of opening: a > 0 opens up, a < 0 opens down
- Find y-intercept: (0, c)
- Plot additional points by choosing x-values on either side of the vertex
- Draw the parabola smoothly through the points
Practice Problems
- Graph f(x) = x² − 6x + 8
- Graph f(x) = −2x² + 4x + 1
- Graph f(x) = 3x² + 12x + 7
- Graph f(x) = −x² + 2x − 3