5 2 Properties of Parabolas Hubarth Algebra II

5 -2 Properties of Parabolas Hubarth Algebra II

The graph of a quadratic function is a U-shaped curve called a parabola. You can fold a parabola so that the two sides match exactly. This property is called symmetry. The fold or line that divides the parabola into two matching halves is called the axis of symmetry. The highest or lowest point of a parabola is its vertex, which is a point on the axis of symmetry.

The vertex is the minimum point or the lowest point of the parabola. The vertex is the maximum point or highest point of the parabola. When the quadratic is in these two forms the vertex are always as follows

Ex 1 Identifying a Vertex Identify the vertex of each graph. Tell whether the vertex is a minimum or a maximum. a. b. The vertex is (1, 2). The vertex is (2, – 4). It is a maximum. It is a minimum.

1 Make a table of values and graph the quadratic function y = 3 x 2. x y = 1 x 2 (x, y) 0 1 (0)2 = 0 3 (0, 0) 2 1 (2)2 = 1 1 3 3 3 1 (3)2 = 3 3 3 (2, 1 1) 3 (3, 3)

Ex 3 Comparing Widths of Parabolas Use the graphs below. Order the quadratic functions (x) = –x 2, (x) = –x 2 – 3 x 2, 1 and (x) = 2 x 2 from widest to narrowest graph. (x) = – 3 x 2 (x) = 1 x 2 2 Of the three graphs, (x) = 12 x 2 is the widest and (x) = – 3 x 2 is the narrowest. So, the order from widest to narrowest is (x) = 1 x 2, 2 (x) = –x 2, (x) = – 3 x 2.

Graph the quadratic functions y = 3 x 2 and y = 3 x 2 – 2. Compare the graphs. x y = 3 x 2 – 2 2 12 10 1 3 1 0 0 2 – 1 2 3 1 12 10 The graph of y = 3 x 2 – 2 has the same shape as the graph of y = 3 x 2, but it is shifted down 2 units.

Graph the function y = 2 x 2 + 4 x – 3. Step 1: Find the equation of the axis of symmetry and the coordinates of the vertex. b – 4 x = – 2 a = 2(2) = – 1 Find the equation of the axis of symmetry. The axis of symmetry is x = – 1. y = 2 x 2 + 4 x – 3 y = 2(– 1)2 + 4(– 1) – 3 = – 5 The vertex is (– 1, – 5). To find the y-coordinate of the vertex, substitute – 1 for x.

Step 2: Find two other points on the graph. Use the 0 and -2 For x = 0, y = – 3, so one point is (0, – 3). The other point is -2 Let x = -2 Find the y-coordinate for x = -2. y = 2(-2)2 + 4(-2) – 3 = -3 For x = -2, y = -3, so another point is (-2, -3).

Step 3: Plot (0, – 3) and (-2, -3) and the vertex (-1, -5) . . .

Step 4 Draw the parabola.

. v(1, 8) Practice . v(3, 0)