2 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 minimumWhen the value of a is a negative number, the parabola opens downward and the vertex is a maximumThe larger the absolute value of a, the steeper (or thinner) the parabola is going to be because the value of y changes more quickly
3 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 vertexthe axis of symmetry is x=hexample: y = 2(x+3)2 +4the vertex is (-3,4) and the axis of symmetry is x= -3the value of a tells us that the parabola opens upwardthe value of a also tells us the “slope” of the parabolathe slope of the parabola is 2/1 – count the number of steps youmust follow in the y-direction and then in the x-direction to getto a point on the parabola that has x and y coordinates that areinteger valuesBoth 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 symmetryThe vertex form also makes is much easier to graph the parabola.
4 example: y = -4(x+8)2 - 6Identify the vertex, axis of symmetry and slope of the parabolathe vertex is (-8,-6) and the axis of symmetry is x= -8the value of a tells us that the parabola opens downwardthe value of a also tells us the “slope” of the parabolathe slope of the parabola is -4/11) y = 3(x - 3)2 + 4Solution: The vertex point is (3, 4); axis of symmetry: x = 32) y = 2(x + 1)2 + 3Solution: The vertex point is (-1, 3); axis of symmetry: x = -13) y = 5(x + 2)2 - 7Solution: The vertex point is (-2, -7); axis of symmetry: x = -2
5 (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
6 The equation is y = 2(x - 2)2 + 1 42-4-224-2The vertex point is (2, 1) The axis of symmetry is at x = 2The slope is 2/1The equation is y = 2(x - 2)2 + 1-4Example: Looking at the graph of the parabola write the equation of the parabola in vertex form