Parabolas and Modeling Section 8.2 Parabolas and Modeling

Objectives Vertical and Horizontal Translations Vertex Form Modeling with Quadratic Functions (Optional)

Vertical and Horizontal Translations The graph of y = x2 is a parabola opening upward with vertex (0, 0). All three graphs have the same shape. y = x2 y = x2 + 1 shifted upward 1 unit y = x2 – 2 shifted downward 2 units Such shifts are called translations because they do not change the shape of the graph only its position

Vertical and Horizontal Translations The graph of y = x2 is a parabola opening upward with vertex (0, 0). y = x2 y = (x – 1)2 Horizontal shift to the right 1 unit

Vertical and Horizontal Translations The graph of y = x2 is a parabola opening upward with vertex (0, 0). y = x2 y = (x + 2)2 Horizontal shift to the left 2 units

Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated 3 units down. The vertex is (0, 3).

Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated left 4 units. The vertex is (4, 0).

Example Sketch the graph of the equation and identify the vertex. Solution The graph is similar to y = x2 except it has been translated down 2 units and right 1 unit. The vertex is (1, 2).

Example Compare the graph of y = f(x) to the graph of y = x2. Then sketch a graph of y = f(x) and y = x2 in the same xy-plane. Solution The graph is translated to the right 2 units and upward 3 units. The vertex for f(x) is (2, 3) and the vertex of y = x2 is (0, 0). The graph opens upward and is wider.

Example Write the vertex form of the parabola with a = 3 and vertex (2, 1). Then express the equation in the form y = ax2 + bx + c. Solution The vertex form of the parabola is where the vertex is (h, k). a = 3, h = 2 and k = 1 To write the equation in y = ax2 + bx + c, do the following:

Example Write each equation in vertex form. Identify the vertex. a. b. Solution a. Because , add and subtract 16 on the right.

Example (cont) b. This equation is slightly different because the leading coefficient is 2 rather than 1. Start by factoring 2 from the first two terms on the right side.