Quadratic Transformations Review

A quadratic function is a polynomial function of degree 2, typically written in the form $$y = ax^2 + bx + c$$ , where $$a$$ , $$b$$ , and $$c$$ are constants and @@a eq 0@@.

Translating a graph vertically means shifting the entire graph up or down along the y-axis. For example, adding a constant to the function shifts it up, while subtracting a constant shifts it down.

A vertical stretch occurs when the value of $$a$$ in the quadratic function $$y = ax^2$$ is greater than 1. This makes the graph narrower.

A vertical shrink occurs when the value of $$a$$ in the quadratic function $$y = ax^2$$ is between 0 and 1. This makes the graph wider.

A horizontal translation shifts the graph left or right. For example, in the function $$y = (x - h)^2$$ , the graph shifts right by $$h$$ units, while $$y = (x + h)^2$$ shifts it left by $$h$$ units.

The vertex is the highest or lowest point of the parabola represented by the quadratic function. It can be found using the formula $$x = -\frac{b}{2a}$$ .

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It can be found using the formula $$x = -\frac{b}{2a}$$ .

The parent function for quadratics is $$y = x^2$$ . It serves as the basic form from which other quadratic functions are derived through transformations.

A negative coefficient reflects the graph of the quadratic function across the x-axis, changing the direction of the parabola.

The direction of a parabola is determined by the sign of the coefficient $$a$$ in the quadratic function $$y = ax^2 + bx + c$$ . If $$a > 0$$ , the parabola opens upwards; if $$a < 0$$ , it opens downwards.