 So to recap, the quadratic graph y equals ax squared plus bx plus c will have a line of symmetry x equals minus b over 2a, have a vertex with x coordinate equal to minus b over 2a. Since the line of symmetry splits the graph, the vertex will be either the lowest point on the graph, if a is greater than 0 and the graph opens upward, or the highest point on the graph if a is less than 0 and the graph opens downward. For example, a thrown object has a height given by this function t seconds after its thrown. When does it reach its highest point and what is its height at that time? So note that y equals h of t will be a quadratic graph because there's a t squared and it opens downward because the coefficient of t squared is negative. And so it will have a highest point. Since that highest point is on the line of symmetry, we find the line of symmetry. And there's one minor change since we're using t instead of x, we'll use t instead of x. And so the line of symmetry will be t equals minus b over 2a. So we'll substitute in our values for b and a and compute. And since this graph opens downward, the highest point on the graph is at the vertex, so it will be where t is approximately 2.041. And so we might say that the object will reach its highest point approximately 2.041 seconds after being thrown. Now since we're given a function for determining the height of the object once we know t to find the height, let t be equal to 2.041. So our function tells us the height, we'll substitute t equal to 2.041 and compute, which tells us the height at that time will be approximately 120.41. Well, not quite. We can't stop here because this number 120.41 is absolutely meaningless, at least until we identify 120.41 what? And since our h of t function tells us the number of meters, this height is 120.41 meters. And it's very important to keep in mind a number by itself is usually meaningless. It's only when you include the appropriate units that the value has significance. Or we might take another example. A coffee shop determines it will sell 5,000 minus 1250x cups of coffee when it charges x dollars per cup. What price will maximize its revenue? Unfortunately, when you apply mathematics to the real world, you have to know something about the real world. In this case, what you have to remember is that revenue is price times the number of items sold. Well, our price is x dollars per cup, and the number of items sold is 5,000 minus 1250x. And so that tells us our revenue is x times 5,000 minus 1250x. We'll expand this and we see that we have a quadratic function. So we see this is a quadratic function with a negative x squared coefficient. So that means that this is a quadratic function which opens downward, and so it will have a maximum value at the vertex. The vertex occurs on the line of symmetry. So we'll find the line of symmetry, which is at 2, and so the vertex is at x equals 2. Now, ordinarily we might want to find the y-coordinate as well, but in this particular case we're looking for the price, and x is the price. And so we might simply say the maximum revenue will occur when the price per cup is 2 dollars.