5-3 Transforming parabolas

The general or standard form of a quadratic function is y = ax2 + bx + c, Another important form of equation used is the vertex form, y = a(x - h)2 + k, where (h, k) is the vertex of the parabola. In both forms, a determines the size and direction of the parabola. When the value of a is a positive number, the parabola opens upward and the vertex is a minimum When the value of a is a negative number, the parabola opens downward and the vertex is a maximum The larger the absolute value of a, the steeper (or thinner) the parabola is going to be because the value of y changes more quickly

When we use the vertex form (y = a(x - h)2 + k ) to graph the parabola the values of h and k are the coordinates of the vertex the axis of symmetry is x=h example: y = 2(x+3)2 +4 the vertex is (-3,4) and the axis of symmetry is x= -3 the value of a tells us that the parabola opens upward the value of a also tells us the “slope” of the parabola the slope of the parabola is 2/1 – count the number of steps you must follow in the y-direction and then in the x-direction to get to a point on the parabola that has x and y coordinates that are integer values Both the vertex form and the standard form give useful information about a parabola. The standard form makes it easy to identify the y-intercept, and the vertex form makes it easy to identify the vertex and the axis of symmetry The vertex form also makes is much easier to graph the parabola.

example: y = -4(x+8)2 - 6 Identify the vertex, axis of symmetry and slope of the parabola the vertex is (-8,-6) and the axis of symmetry is x= -8 the value of a tells us that the parabola opens downward the value of a also tells us the “slope” of the parabola the slope of the parabola is -4/1 1)  y = 3(x - 3)2 + 4 Solution:  The vertex point is (3, 4); axis of symmetry: x = 3 2)  y = 2(x + 1)2 + 3 Solution:  The vertex point is (-1, 3); axis of symmetry: x = -1 3)  y = 5(x + 2)2 - 7 Solution:  The vertex point is (-2, -7); axis of symmetry: x = -2

(3/2,0) (1,1) (2,1) (0,9) (3,9) To find a in the equation y = a(x - h)2 + k, substitute the points of the vertex in for h and k, then pick one set of coordinates and substitute them in for x and y. Solve for a to find the slope of the parabola

The equation is y = 2(x - 2)2 + 1 4 2 -4 -2 2 4 -2 The vertex point is (2, 1) The axis of symmetry is at x = 2 The slope is 2/1 The equation is y = 2(x - 2)2 + 1 -4 Example: Looking at the graph of the parabola write the equation of the parabola in vertex form