The family of quadratic functions

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This lesson will summarize the family of quadratic functions using the parent function y = x2

Vertical stretch or shrink, and/or reflection in x-axis

Parent function: y = x2

Reflection in x-axis: y = -x2

Stretch by a factor of a: y = ax2

If a > 1, then the graph will stretch or become more narrow than the graph of y = x2

Shrink by a factor of a: y = ax2

If 0 < a < 1, then the graph will shrink or become more wide than the graph of y = x2

(Stretch or Shrink) and reflection in x-axis: y = -ax2

The figure below shows what the graph will look like for a stretch, shrink, or a reflection using y = x2 as the parent function. Notice that the blue graph is the parent function or y = x2.

Family of quadratic functions

Vertex form

Parent function: y = x2

Vertex form: y = a(x - h)2 + k

The vertex is (h, k) and the line x = h is the axis of symmetry.

The graph and vertex of y = ax2 shifts h units horizontally and k units vertically.

For k > 0, the graph shifts up

For k < 0, the graph shifts down

For h > 0, the graph shifts to the right

For h < 0, the graph shifts to the left

The figure below shows what the graph will look like for horizontal and vertical shifts using y = x2 as the parent function. Notice that the blue graph is the parent function or y = x2.

Family of quadratic functions