# Yahoo Answers 06-16-14

Topic:
Parabola
The vertex form of quadratic equation is given by y=a(x-h)^2+k where a, h and are constants and a not equal to 0. Discuss the change of the shape and position of the graph if (a) a varies from -2 to 2 when h=3 and k=4. (b) h varies from -2 to 2 when a=2 and k=4 (c) k varies from -2 to 2 when a=2 and h=4 The vertex form comes from solving the standard form by completing the square. So you are going from y=Ax^2+Bx+C to y=a(x-h)^2+k y=4x^2-8x+1 factor out 4 y=4(x^2-2x+___)+1 1/2 of -2 is -1, squared is 1 we are adding 1 inside the parentheses which is really 4 since we are multiplying by 4. to keep the equation the same we have to subtract 4 y=4(x^2-2x+1)+1-4 y=4(x^2-2x+1)-3 y=4(x-1)^2-3 a=4, h=1, k=-3 the minus sign in the binomial is part of the formula so if it is x-h, h is positive, if it&#39;s x+h, h is negative k behaves normally. +k is positive, -k is negative the vertex is at (h,k) a is a scalar if |a|>1 the parabola is more narrow as a gets bigger if |a|<1 the parabola is flatter as a gets smaller if a is positive the parabola opens upward if a is negative the parabola opens downward (if a=0 you have a horizontal line y=k) if a>0 and k>0, the parabola has no real roots (it opens upward and the vertex is above the x-axis, it will not cross the x-axis) or if a<0 and k<0 it has no real roots (it opens downward and the vertex is below the x-axis) if k=0 the vertex is on the x-axis and it has one root if a and k have different signs, the parabola has two real roots To answer the questions, a) as a goes from -2 to 2... the vertex is fixed at (3,4) it starts as a narrower parabola pointing downward, it will get wider and wider until a=0 and it's a horizontal line then the parabola will open pointing upward and get narrower and narrower b) as h goes from -2 to 2 this is an upward facing parabola (a bit narrow) since a=2 the vertex will start at (-2,4) and move to the RIGHT until it stops at (2,4) NOTE: the equation will start at y=2(x+2)^2+4 and stop at y=2(x-2)^2+4 c) as k goes from -2 to 2 again it is narrow and upward facing the vertex will start at (2,-2) and move UP until it stops at (2,2)