4.1 Quadratic Functions and Transformations A parabola is the graph of a quadratic function, which you can write in the form f(x) = ax 2 + bx + c, where a ≠ 0. The graph of any quadratic function is a transformation of the graph of the parent quadratic function, y = x 2. The vertex form of a quadratic function is f(x) = a(x – h) 2 + k, where a ≠ 0. The axis of symmetry is a line that divides the parabola into two mirror images. The equation of the axis of symmetry is x = h. The vertex of the parabola is (h, k), the intersection of the parabola and its axis of symmetry.

Graphing a Function of the Form f(x) = ax 2 What is the graph of ?

Graphing Translations of f(x) = x 2 Graph each function. How is each graph a translation of f(x) = x 2 ? A.g(x) = x 2 – 5 Translate the graph of f down 5 units to get the graph of g(x) = x 2 – 5.

Interpreting Vertex Form For y = 3(x – 4) 2 – 2, what are the vertex, the axis of symmetry, the maximum or minimum value, the domain and the range? y = a(x – h) 2 + k y = 3(x – 4) 2 – 2 The vertex is (h, k) = (4, -2) The axis of symmetry is x = h, or x = 4. Since a > 0, the parabola opens upward. k = -2 is the minimum value. Domain: All real numbers. There is no restriction on the value of x. Range: All real numbers ≥ -2, since the minimum value of the function is -2.

Using the Vertex Form What is the graph of f(x) = -2(x – 1) 2 + 3? a = -2  the parabola opens downward. h = 1, k = 3 vertex is (1, 3) axis of symmetry: x = 1

More Practice!!!!! Homework – Textbook p. 199 #7 – 37.