 In a kind and gentle universe, we're always going to be given the exact formula for a function. We don't live in that universe, so we often need to construct the formula from given information. So let's go to Scotland where they practice the fine art of caber tossing. That's basically throwing a telephone pole. So let's say a tosser throws a caber. The center of the caber describes a parabola and reaches a maximum height of 20 feet. The center of the caber lands at a distance of 30 feet from the tosser. Let's write an equation for the path of the center of the caber, and we'll assume the tosser's position corresponds to the initial location of the center of the caber. So let's make our tosser's position the origin. If our horizontal and vertical distances are measured in feet, then we know that the parabola passes through the tosser's position at 0, 0. It also passes through the landing point at 30, 0. The maximum height of the parabolic path occurs at the vertex. Now remember the line of symmetry passes through the vertex, and since the parabola has points at 0, 0 and 30, 0, the line of symmetry must be midway between these points. And so the line of symmetry is x equal to 15. Since the line of symmetry passes through the vertex, then we know the vertex has coordinates x equal to 15, and y equal to 20, the maximum height. And so we know the vertex is at 15, 20. Now the equation for a parabola with vertex hk is y equal to a x minus h squared plus k, and we know the vertex. So we know h equals 15, k equals 20. And so the equation is y equals something x minus 15 squared plus 20. Since the parabola also passes through 0, 0, we also know that x equals 0, y equals 0 makes the equation true. So substituting in these values, we get an equation where a is the only value we don't know. So solving for a gives us a equals minus 20 to 25th, and that's the equation of our parabola. Or let's find the x and y intercepts of the parabola with the equation y equals a x squared plus 6x plus c, where we know where the vertex is located. Since we know the vertex, we could use the formula y equals a x minus h squared plus k. But this throws away the fact that we have the formula y equals a x squared plus 6x plus c. So let's see if we can use that formula. So remember the graph of y equals a x squared plus b x plus c is a parabola with line of symmetry x equals minus b over 2a. And remember the line of symmetry passes through the vertex 3 negative 10. Since this has x equals 3 and equals means replaceable, we can use this relationship to find a. And we see that a is equal to negative 1. And so we find our equation y equals minus x squared plus 6x plus something. Since the parabola passes through the point 3 negative 10, we know that x equal to 3, y equal to negative 10 makes the equation true. And we can use this to find our value of c. So equals means replaceable, so we'll replace x with 3 and y with negative 10 and solve for c, which gives us our equation. Now remember the x intercepts are where y is equal to 0. So if y is equal to 0, we get the equation. Pick your favorite method of solving this. How about the quadratic formula? And we see that we get a complex number as a solution. So there is no x intercept. Actually, we should have known that from the beginning. a is equal to negative 1, so the parabola opens downward and the vertex at 3 negative 10 is below the x-axis. So if you think about what that looks like, there's no way that this parabola can intersect the x-axis. The y-intercept is where x is equal to 0. So equals means replaceable, we'll replace x with 0 and find y is equal to negative 19, and so the y-intercept is at 0 negative 19.