 So last time we found the graph of y equals a x squared will be a parabola opening upward if a is greater than zero and Opening downward if a is less than zero But that was for a drastically simplified second-degree polynomial. What about y equals a x squared plus b x plus c? We'll need to do a little hand-waving here So we might approach this as follows since our x value is being squared then a x squared will Generally be larger than bx and c and so the shape of the graph is mostly determined by the a x squared term And so consequently for a not equal to zero the graph of y equals a x squared plus b x plus c is a Parabola where if a is greater than zero the parabola opens upward and if a is less than zero the parabola opens downward So let's try to sketch the graph of y equals x squared minus 18x minus 30 And here's an important idea when the word sketch the graph shows up sketch means to show the general features of the graph So what does that mean? Well at this point the only thing we really know about the shape of the graph is coming from our theorem Since the equation is in the form y equals a x squared plus b x plus c Then we know it will have the shape of a parabola and since a is greater than zero it will open upward So it generally looks something like this However, there are some details which we have to work out So let's look at some further graph features the graph of y equals a x squared can be viewed as having two parts The part that's to the right of the y axis and the part that's to the left of the y axis But the two parts look exactly the same We say that the y axis is a line of symmetry for the graph of y equals a x squared Now the line of symmetry comes from the quadratic formula Suppose some point h k is on the graph of y equals a x squared plus b x plus c Then we know that x equals h is a solution to k equals a x squared plus b x plus c We get a second solution from the quadratic formula But notice that our two solutions will be x equal to minus b over 2a plus some amount or x equals minus b over 2a minus the same amount and this means that the two points are right of the line x equals minus b over 2a by some amount and Left of the line x equals minus b over 2a by the same amount So x equals minus b over 2a splits the graph and this leads to the following idea The graph of y equals a x squared plus b x plus c has the line of symmetry x equals Minus b over 2a This allows us to introduce an important point since the line of symmetry splits a quadratic graph The point on the line of symmetry has a special property if the graph opens upward This point is the lowest point on the graph On the other hand if the graph opens downward this point is the highest point on the graph We call this point the vertex So for example, let's find the line of symmetry of y equals 4x squared minus 8x plus 12 and the vertex and then sketch the graph So our theorem says that the line of symmetry of the graph y equals x squared plus bx plus c is going to be x equals minus b over 2a So we'll fill in those values for our graph b is the coefficient of x minus 8 a is the coefficient of x squared that's 4 and so our line of symmetry is going to be x equal to 1 So if we want to sketch the graph we'll draw the line x equals 1 Now the vertex is on the line of symmetry So x equals 1 and y is given by our formula So we find that y is equal to 8 and so the vertex will be at 1 8 and Finally our theorem says that if our coefficient of x squared is greater than 0 our parabola opens upward So we know the line of symmetry We know the vertex and so our parabola must look something like this