3.3 The Inverse of a Quadratic Function Given the function y = x2, what is the inverse function? x square it y x y 4 square it 16 16 4 – 4 square it 16 16 4 In order to obtain – 4 as part of the inverse, we must include y = and y = .

1- Sketch the graph of y = x2 3- Sketch the inverse of y = x2

The inverse of y = x2 is not a function. It does not pass the vertical line test One way to assure that the inverse will be a function is to limit the domain of y = x2.

Example: Graph the relation y = x2, x ³ 0. Equation of the inverse is The graph of is a function. Domain: x ³ 0 Range: y ³ 0

Graph the relation y = x2, x £ 0. Equation of the inverse is The graph of is a function. Domain: x ³ 0 Range: y £ 0

Determine the equation of the inverse. Example: How can I restrict the domain of the function so that the inverse will be a function? y = (x + 2)2 + 1 The vertex is (– 2, 1) Domain: x ³ – 2 Range: y ³ 1 Determine the equation of the inverse.

Example: Determine the vertex of the parabola The method is called completing the square. y = 2x2 + 8x – 1 Factor the coefficient of x2 from the first two terms. Complete the square Remove the last term in parentheses Factor Vertex is (– 2, – 9)

Sketch the graph of y = 2(x + 2)2 – 9 Domain: How can we restrict the domain so that the inverse will be a function? Domain:

Determine the equation of the inverse of: y = 2(x + 2)2 – 9

Example: Determine the vertex of the parabola The method is called completing the square. y = – x2 + 6x + 2 Factor the coefficient of x2 from the first two terms. Complete the square Remove the last term in parentheses Factor vertex is (3, 11)

Determine the equation of the inverse function Determine the equation of the inverse function. Restrict the range of the inverse function so that it is also a function.