My problem says to graph the inequality y ? (x-3)^2 +2
So I worked the problem out and got to y ? x^2 - 6x + 11
Now where I am is stuck is how do I begin to graph this.
Do you recognize that this is the graph of a parabola? The original equation (forget the inequality part for a moment) is in “vertex form.” The vertex can be read right out of the equation and is the point (3,2).
The vertex is always on the axis of symmetry. The axis of symmetry is the line x = 3.
The form you converted it into is “standard form.” In standard form, we can easily see that the y-intercept is 11 (just substitute 0 in for x).
You should be able to graph the parabola now.
The inequality means that either the inside or the outside of the parabola is the region we’re talking about. To decide which it is, just pick some arbitrary point either inside or outside the parabola. Plug the x and y values of that point into your inequality and see if it is true or not. The region you want is the region containing points that make the inequality true.
In this particular case, the parabola opens upward, like a U. For a more visual way to understand which region the inequality is defining, just place your pencil on any point on the parabola. Now examine the inequality: it says “y is less than,” meaning we would move our pencil down from the point on the parabola – and we find we’re in the region “outside” the parabola. Shade this region. Make sense?
