if the parabola does not have any x intercepts can you still find the interval increase and decrease.

Question

here's the problem: y=5x^2-10+20. Please help!

Answer 1

Yes, you can!  If you look at any parabola, you'll find that the only point on the graph where there is no change in y-values is the vertex.  The vertex is the transition point from one interval of movement to the other (i.e. from the interval of increase to the interval of decrease or from the interval of decrease to the interval of increase).  The part of the vertex that we are interested in is the x-coordinate.  In order to find the x-coordinate of the vertex, you need to put the quadratic function in vertex form.  If your function is y = ax^2 + bx + c, then vertex form will be y = a(x + b/(2a))^2 + c - b^2/(4a).  From here, you can read off the x-coordinate of the vertex.  It is simply the negation of the constant term inside the parentheses associated with the x term (-b/(2a)).  The next piece of information we need to know is whether the parabola opens up or down.  This can be determined by the leading coefficient, a.  If a > 0, then the parabola opens up.  If a < 0, then the parabola opens down.  In this case, the leading coefficient is 5, so the parabola opens up.  This implies that the graph of the parabola decreases to the left of the vertex and increases to the right of the vertex.  If the parabola opened down, then it would be the opposite.  Therefore, the interval of decrease is (-infinity,-b/(2a)) and the interval of increase is (-b/(2a),+infinity).