Transformations of Quadratics

The parent function of a quadratic equation is $$f(x) = x^2$$ .

To translate a function to the left by 'h' units, replace 'x' with 'x + h' in the function's equation.

The vertex form of a quadratic function is $$f(x) = a(x - h)^2 + k$$ , where (h, k) is the vertex.

To find the y-intercept, set $$x = 0$$ in the function's equation and solve for $$y$$ .

The 'a' value determines the direction of the parabola (upward if a > 0, downward if a < 0) and its width.

A negative 'a' value reflects the parabola over the x-axis.

The range of a quadratic function is determined by the vertex; if the parabola opens upwards, the range is $$[k, \, \, \text{infinity}]$$ , and if it opens downwards, it is $$(-\text{infinity}, k]$$ .

The vertex is the highest or lowest point of the parabola, depending on its direction.

To reflect a quadratic function over the x-axis, multiply the entire function by -1.

To translate a function down by 'k' units, subtract 'k' from the function's equation.