In this video, we introduce quadratic expressions, and illustrate how they can be understood visually and geometrically, using a fundamental class of curves known as parabolas. Recall from the last module, that we discussed equations of lines, one of the main forms of which is the slope intercept equation, which has the form y equals mx plus k, where m is the slope and k is the y-intercept. The combination of symbols on the right-hand side, mx plus k, is called a linear expression, and it is formed with just one multiplication and one addition, which when you think about it is very rudimentary arithmetic, which is the whole point of equations of lines. They are intended to be simple and easy to handle. Let's gradually increase the complexity. By adding a further term involving the square of x and another constant, so there are three constants renamed alphabetically a, b, and c, to get what is called a quadratic expression of the general form y equals ax squared plus bx plus c. Remember a, b, c are constants and x, y are variables. To be what we might call a proper quadratic, we want a to be non-zero for otherwise, the expression reverts to just bx plus c, the linear case which we've already considered. You might be curious the prefix quad in quadratic comes about because there are four sides to a square and x squared of course represents the area of a square of side length x. Indeed the simplest case is just y equals x squared. When a is equal to one and b and c are both equal to zero. What might the relationship look like between x and y equals x squared when we consider or plot points in the x-y plane? Let's look at some simple values. When x equals 0, then y is zero squared, which equals 0, so we get the point (0,0) which is just the origin. When x equals 1, we get y equals 1 squared equal to 1, so we get the point (1,1). When x equals minus 1, we get the same square value plus 1 and now the point (-1,1). When x equals 2 or minus 2, we get the same square value plus 4, so you get the points (2,4) and (-2,4). We can plot these five points on the Cartesian x-y plane, and join the points using a smooth curve. What we've produced is another very important curve in the history and development of mathematics known as a parabola. Let's look at some variations. The simplest example, which was just discussed, arises when a equals 1, b equals c equals 0. If we take a equals 1 and b equals 0 again but take c equal to 2, we get y equals x squared plus 2. For enough variation, we could take a equals 1, b equals 4 and c equals 4, to get y equals x squared plus 4x plus 4. Let's sample some points (x,y) for each of these curves, where the values we input for x are 0, 1, minus 1, 2 and minus 2. Here are the five points that we obtained earlier for y equals x squared, and now points for y equals x squared plus 2, you can quickly check by adding 2 to the y values in the previous list. And also points for y equals x squared plus 4x plus 4. The values for the third example might look haphazard. But let's see what actually happens when we plot the points on the x-y plane in each case and join the points with smooth curves. Here's the parabola for y equals x squared that we found before. The second quadratic y equals x squared plus 2 simply shifts these y-values upwards by two units, and so the curve joining these new points is just the original parabola shifted upwards by two units. To plot the five points for the third quadratic, we need some extra room, which we get by halving the scale on the x and y axis from the diagrams we used before, and you can easily check that we also get these other points. When x equals minus 3, minus 4, minus 5, and minus 6, and then join up the points with a smooth curve, which again turns out to be a parabola. If you include the original parabola for y equals x squared, that's the blue curve on the same diagram, you discover that the pink curve is obtained by shifting the blue curve uniformly two units to the left. Now how can that be? The rules for the quadratics looks so different. Observe that, in fact, x squared plus 4x plus 4 is a perfect square. In the sense that it is the square of x plus 2, which is really just x squared with x replaced by x plus 2, so that to get the y-values for the pink curve, all one needs to do is input x values that are two units less than those feeding into the blue curve, and the shifting phenomenon is explained. In fact, every curve described by the general quadratic y equals ax squared plus bx plus c is a parabola, and just the result of dilation and shifting up and down or back and forth, the simplest parabola associated with y equals x squared. This completely general fact, is not obvious, but it's explained carefully in the notes, together with some principles regarding dilations and shifts of general curves in the x-y plane. If the constant a in the quadratic expression happens to be negative, then the dilation effect includes also flipping the parabola upside down. We often say by reflecting in the x-axis. For example, consider the parabola for y equals minus x squared where a is equal to minus one and b and c are both zero, and we get an upside down parabola. In fact, the result of reflecting the parabola for y equals x squared in the horizontal x-axis. Bowl-shape up becomes bowl-shape down, using any horizontal reflection. Notice that the apex of the parabola is at the origin. The apex turns out to be important for applications. Here's a trickier one, y equals 2x minus x squared, just rearranged slightly to emphasize the minus x squared term, and we know it must yield an upside down parabola, because of the minus sign. But where does the parabola lie on the x-y plane? Let's do some algebraic manipulation. First write y as the negative of x squared minus 2x, all in brackets, then use the technique of completing the square, by adding and subtracting one inside the brackets, as this doesn't change the overall value of the expression. Bring the minus minus 1 outside the brackets as plus 1, and then recognize the bracketed expression as the perfect square, x minus 1 squared. So finally, we've written y as minus x minus 1, all squared, plus 1. To understand the curve, because what we've just done, we can relate this in a natural way to the simplest upside down parabola of all. Start with minus x squared, replace x by x minus 1, and then add 1, and you recover the expression minus x minus 1, all squared, plus 1. You can see the visual geometric effect of these two algebraic steps. Start off with the upside down parabola y equals minus x squared. Take a copy of the parabola, and watch the effect of shifting one unit to the right, corresponding algebraically to replacing x by x minus 1, followed by moving one unit vertically upwards, corresponding algebraically to adding plus 1. You can see that we recover the pink parabola from the green one. Notice how the apex of the green curve is at the origin (0,0), and this is shifted to the point (1,1), which becomes the apex of the pink curve. Notice also how the pink curve crosses the x-axis at x equals 0 and x equals 2. This makes perfectly good sense since this occurs when y is equal to 0. If we go back to the original expression, instead of completing the square, if we put y equal to 0, then we get a factorization of zero in just one step, x times 2 minus x. And remember in an earlier video, we said that a factorization of zero is very useful, leading to x equals 0 or x equals 2, where the curve crosses the x-axis. There are lots of different ways of looking at these curves, with experience, you would realize that one of the fastest ways to visualize examples like the parabola y equals 2x minus x squared, is to recognize that it's upside down, as the coefficient of x squared is negative, and crosses the x-axis at values dictated by a factorization of zero. There's much more that can be said about parabolas especially about the apex, the point where it turns around. To get a more complete story, we'll introduce and discuss the quadratic formula in the next video. We've made a good beginning here and covered a lot of ground, we've introduced general quadratic expressions, shown how they correspond to important curves known as parabolas, and illustrated how different parabolas relate to each other, especially by shifting and turning things upside down, by reflecting in the x-axis. Please read the notes and when you're ready, please attempt the exercises. Thank you very much for watching, and I look forward to seeing you again soon.
