Lecture Notes: Graphing Quadratic Functions

Overview

• Focus on graphing quadratic functions in detail.
• Example: Graphing ( (x-1)^2 - 2 ).

Quadratic Function Standard Form

• Standard form: ( a(x-h)^2 + k ).
• Given example fits the standard form:
  • ( a = 1 ) (since no coefficient is present, it's understood to be 1).
  • ( a > 0 ), thus the parabola opens upwards.
  • Vertex ( (h, k) ) is given by ( (1, -2) ).

Graphing the Equation

1. Identifying the Vertex:

   • Vertex is at ( (1, -2) ).
   • Mark the vertex on the graph.

2. Axis of Symmetry:

   • Equation of axis of symmetry: ( x = 1 ).
   • Passes through the vertex at ( x = 1 ).

3. Determining Width of Parabola:

   • Choose a point on either side of the axis of symmetry.
   • Example: Choose ( x = 2 ).
     • Calculate ( y ): ( (2-1)^2 - 2 = -1 ).
     • Point: ( (2, -1) ).
   • Reflect point across axis of symmetry to find another point.

4. Sketching the Parabola:

   • Plot the points and sketch the parabola.

Domain and Range

• Domain:
  • All real numbers ((-\infty, \infty)).
  • Parabola opens out infinitely without gaps.
• Range:
  • Starts from vertex y-value to infinity: ([-2, \infty)).
  • Lowest y-value is (-2) at the vertex.](streamdown:incomplete-link)

Summary of Steps

• Determined the direction of opening from ( a ).
• Found and plotted the vertex.
• Drew the axis of symmetry.
• Determined the width by plotting another point.
• Stated domain and range.
• Equation of axis of symmetry: vertical line, ( x = 1 ).

Conclusion

• Vertex: ( (1, -2) ).
• Axis of Symmetry: ( x = 1 ).
• Domain: All real numbers.
• Range: ( y \geq -2 ).

• Next video will include another example of graphing a quadratic function.