4.1 Quadratic Functions and Transformations• A parabola is the graph of a quadratic function, which

you can write in the form f(x) = ax2 + bx + c, where a ≠ 0.

• The graph of any quadratic function is a transformation of the graph of the parent quadratic function, y = x2 .

• 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) = ax2 • What is the graph of ?21

( )2

f x x

Graphing Translations of f(x) = x2 • Graph each function. How is each graph a translation

of f(x) = x2 ?

A.g(x) = x2 – 5

Translate the graph of f down 5 units to get the graph of g(x) = x2 – 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 + ky = 3(x – 4)2 – 2The 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.