Transformations of Quadratic and Absolute Value Functions

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

A negative coefficient indicates that the graph of the function is reflected over the x-axis.

The parent function of absolute value is $$f(x) = |x|$$ .

This equation represents a horizontal shift 3 units to the right and a vertical shift 2 units up.

Adding a constant outside the absolute value function results in a vertical shift of the graph.

The standard form of a quadratic function is given by $$y = ax^2 + bx + c$$ .

The axis of symmetry can be found using the formula $$x = -\frac{b}{2a}$$ from the standard form $$y = ax^2 + bx + c$$ .

This represents a horizontal shift 2 units to the left of the parent function $$f(x) = x^2$$ .

The coefficient 'a' determines the direction of the parabola (upward if a > 0, downward if a < 0) and its width (narrower if |a| > 1, wider if |a| < 1).

This represents a vertical stretch by a factor of 3, a reflection over the x-axis, a horizontal shift 1 unit to the right, and a vertical shift 5 units up.