Graphing Quadratic Functions in Vertex Form Using Transformations - Algebra-2

1. Fundamental Concepts

Parabola: The graph of a quadratic function, which is a U-shaped curve.
Vertex: The key point of a parabola, which is the minimum or maximum point.
Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
Vertex Form of a Quadratic Function: The form is , where is the vertex of the parabola.

Role of a:
- Responsible for vertical compression or stretch of the parabola. If , the parabola is stretched vertically; if , it is compressed vertically.
- The sign of a determines the direction the parabola opens. If , the parabola opens upward; if , it opens downward.
Role of h: Causes horizontal translation of the parent function . If , the graph shifts h units to the right; if , the graph shifts units to the left.
Role of k: Causes vertical translation of the parent function . If , the graph shifts k units upward; if , the graph shifts units downward.
Transformation Rules: When graphing a quadratic function in vertex form, we start with the parent function (which has a vertex at and opens upward) and apply the transformations based on the values of a, h, and k in the vertex form .

Simple: Graph $f(x)=(x-3)^2 + 4$
- Start with the parent function with vertex .
- Since , shift the graph 3 units to the right.
- Since , shift the resulting graph 4 units upward.
- The vertex of the new parabola is , and it opens upward (since ) with the same width as the parent function.
Medium: Graph $f(x)=2(x + 3)^2$
- Begin with (vertex ).
- Because , shift the graph 3 units to the left.
- Due to, stretch the graph vertically by a factor of 2.
- The vertex is , and the parabola opens upward, being narrower than the parent function.
Difficult: Graph $f(x)=-3(x + 1)^2-2$
- Start with (vertex ).
- Since , shift the graph 1 unit to the left.
- Because , reflect the graph over the x-axis and stretch it vertically by a factor of 3.
- Due to , shift the resulting graph 2 units downward.
- The vertex is , and the parabola opens downward, being narrower than the parent function.

Identify the Vertex: From the vertex form , directly obtain the vertex .
Determine the Direction of Opening: Check the sign of a to know if the parabola opens upward () or downward ().
Apply Transformations Step by Step: Start with the parent function and apply horizontal translation (based on h), vertical stretch/compression and reflection (based on a), and vertical translation (based on k) in sequence.
Plot Key Points: After determining the vertex, use the symmetry of the parabola (with respect to the axis of symmetry ) to plot additional points. For example, if one point is on the parabola, then the point (symmetrical to about the axis of symmetry) is also on the parabola.
Sketch the Parabola: Connect the plotted points smoothly to form the parabola.