Transformations of the parent quadratic function, f(x) = x², are directly embedded within the vertex form of a quadratic function, f(x) = a(x - h)² + k. The parameters a, h, and k dictate specific transformations of the graph.
Here's how each parameter relates to transformations:
-
'a' (Vertical Stretch/Compression and Reflection): The 'a' value controls the vertical stretch or compression and any reflection across the x-axis.
- If |a| > 1, the graph is vertically stretched (narrower).
- If 0 < |a| < 1, the graph is vertically compressed (wider).
- If a > 0, the parabola opens upwards.
- If a < 0, the parabola opens downwards (reflected across the x-axis).
-
'h' (Horizontal Translation): The 'h' value controls the horizontal translation (shift) of the graph.
- If h > 0, the graph is shifted h units to the right. Remember the negative sign in the vertex form: (x - h).
- If h < 0, the graph is shifted |h| units to the left. This is because (x - (-h)) becomes (x + h).
-
'k' (Vertical Translation): The 'k' value controls the vertical translation (shift) of the graph.
- If k > 0, the graph is shifted k units upwards.
- If k < 0, the graph is shifted |k| units downwards.
In Summary:
The vertex form, f(x) = a(x - h)² + k, encapsulates transformations by providing direct access to the parameters that cause:
- Vertical Stretch/Compression/Reflection: Controlled by 'a'
- Horizontal Translation: Controlled by 'h'
- Vertical Translation: Controlled by 'k'
The vertex of the parabola is located at the point (h, k), demonstrating how h and k directly translate the parent function's vertex (0, 0) to a new location.
