In this video, we’re gonna look at some graphs of quadratic functions and find links between what they look like, and the various coefficients in their equations. We’ll explore why those links exist, by considering the quadratic formula.
Remember, the quadratic equation has got an 𝑥 squared term, an 𝑥-term, and a constant term. So that’s some number times 𝑥 squared plus or minus some number times 𝑥 plus or minus some constant number on the end.
So for example 𝑦 equals three 𝑥 squared plus two 𝑥 minus five. In that case, the 𝑎-value is positive, it’s positive three. The 𝑏-value is positive, it’s positive two. But the 𝑐-value was negative, it was negative five.
Another example is 𝑦 equals negative two 𝑥 squared. So in this case, the 𝑎-value, the coefficient of 𝑥 squared, is negative two. But the coefficients of the 𝑥-term and the constant are just series of 𝑏 equals zero and 𝑐 equals zero.
And one more example, 𝑦 equals a quarter 𝑥 squared plus two-fifths. In that case, the 𝑎-value will be a quarter, the 𝑏-value will be zero, and the 𝑐-value will be two-fifths.
So 𝑏 or 𝑐 could be equal to zero, but to be a quadratic equation, the value of 𝑎 could never be zero.
If we plot any quadratic function on a graph, we’ll always get a symmetrical parabola like one of these two, either a U-shape or an upside-downed U-shape. Now you probably remember that the value of 𝑎, the coefficient of 𝑥 squared that is, tells you about how wide or thin the curve is, and which way up it is, remember, positive happy smiley curves or negative sad down-faced curves. And changing the 𝑥-coefficient, so the value of 𝑏 moves the curve left or right on the graph. And the 𝑐-value, the constant term, tells you the value of the function when the 𝑥-input is zero. In other words, where it cuts the 𝑦-axis or the 𝑦-intercept.
So looking at the coefficients and the constant term can tell us a lot about what the graph of the function would look like. And the other way around, if we see the graph, we can tell what the coe- some of the coefficients are gonna be. But there’s also another aspect we need to know about. Where does the curve cut the 𝑥-axis? In other words, which 𝑥-inputs generate 𝑦-outputs of zero?
Now you probably spent quite a long time working out such things, maybe by reading values from graphs, so using systematic trial and improvement. Perhaps factoring, or using the quadratic formula, or even by completing the square. But you may have noticed that sometimes you get two answers, sometimes you get one, and sometimes you don’t get any. Maybe the quadratic expression can’t be factored, or maybe the formula just goes wrong and says math error on your calculator, when you type it in.
Now some quadratics have two roots and that’s because they cut the 𝑥-axis in two places. So there are two 𝑥-values that generate 𝑦-coordinates of zero. Some have one root. For what we call repeated roots, that’s two roots, but they just happened to be in the same place. And others have no roots, well no real roots. There is a way of using things called imaginary or complex numbers to generate nonreal roots. But we’re not going to worry about that yet. So if the curve turns around and heads back up, or maybe turns around and heads back down depending on whether they’re coming from above or below, without crossing the 𝑥-axis anywhere, then we say there are no roots. And that’s because there are no points on that curve that have a 𝑦-coordinate of zero. None of the 𝑥-inputs generate a 𝑦-coordinate of zero.
Let’s have a good look at that quadratic formula then. The solutions of 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equal zero, where 𝑎 is not equal to zero, are given by 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.
So the idea is you take your quadratic equation and you just plug in the 𝑎, 𝑏 and 𝑐 values. And that’ll tell you the 𝑥-coordinates for which the 𝑦-coordinate is zero. But whether there are two solutions, one solution, or no solutions, all hinges on this little bit here; and it’s called the discriminant. So when using the quadratic formula, we need to find the square root of the discriminant in order to help us find the 𝑥-coordinates of the points where the curve cuts the 𝑥-axis. Now that’s the issue. So if the discriminant is positive, we’re gonna be finding the square root of a positive number. And that will give us two different values, a positive version and a negative version, so will generate two values of 𝑥. If the discriminant is equal to zero, then we’re gonna be taking the square root of zero, which is zero. So that’s only one value, so we’re gonna find just one value. And if the discriminant is negative, we’re gonna be trying to find the square root of a negative number, which you know that’s gonna be very difficult indeed, if not impossible; unless you invent an entire new system of numbers called imaginary numbers.
Let’s look at a couple of examples then. So if 𝑦 equals 𝑥 squared plus three 𝑥 plus two, that means 𝑎 is one, 𝑏 is three and 𝑐 is two. So if we put the 𝑦-coordinate equal to zero so that we can find out where it cuts the 𝑥-axis, if we plug all those numbers into the quadratic formula, then the discriminant here is three squared. That’s nine minus four times one times two; that’s eight. So nine minus eight is positive one. And the square root of one could be positive one, could be negative one. Positive one times positive one is one, negative one times negative one is also one. So that generates two possible solutions. So for this particular quadratic, an 𝑥-coordinate of negative one generates a 𝑦-coordinate of zero, or an 𝑥-coordinate of negative two generates a 𝑦-coordinate of zero. In other words, it cuts the 𝑥-axis in two places when 𝑥 is equal to negative one and when 𝑥 is equal to negative two. So when the discriminant 𝑏 squared minus four 𝑎𝑐 was bigger than zero, we had two real roots to our equations; two 𝑥-values that generate a 𝑦-coordinate of zero.
Okay. Let’s look at another example then. 𝑦 equals 𝑥 squared plus two 𝑥 plus one. So in this quadratic 𝑎 is equal to one, 𝑏 is equal to two, and 𝑐 is equal to one. So if we put 𝑦 equal to zero to try to find the 𝑥-coordinates where it cuts the 𝑥-axis, and when we plug those numbers into our quadratic equation, this bit inside the square root here, the discriminant, turns out to be four minus four, which is zero. So when we go on to solve that equation, the values of 𝑥 are gonna be negative two plus or minus the square root of zero. So obviously the square root of zero is zero. So we’re adding zero to negative two and we’re subtracting zero from negative two. So clearly, our two solutions are gonna be exactly the same, in this case, negative one. So basically, an 𝑥-coordinate of negative one generates a 𝑦-coordinate of zero. But there aren’t any other 𝑥-coordinates to do that, so this is a curve which just touches the 𝑥-axis in one place.
So let’s have a look at one more example then. 𝑦 equals two 𝑥 squared plus 𝑥 plus three. So now 𝑎 is two, 𝑏 is one and 𝑐 is three. And plugging those numbers into our quadratic formula gives us a discriminant of one squared minus four times two times three; that’s negative twenty-three. And when we try to solve this now, we’ve gotta find the square root of negative twenty-three. But you can’t get a square root of a negative number because if I take a number multiplied by itself, whether it’s positive or negative, I’m always gonna get a positive answer.
So this is an example of a quadratic that’s got no roots. In other words, there aren’t any 𝑥-coordinates which generate a 𝑦-coordinate of zero. We can’t find any real values of 𝑥, which are gonna enable us to calculate this square root of a negative number here. So we can use this information, just by analyzing the discriminant, to tell whether there are gonna be two roots, one root, or no roots for our quadratic equation. If 𝑏 squared minus four 𝑎𝑐 is bigger than zero, then there’re gonna be two roots. So you could just work out the value of 𝑏 squared, the value of four 𝑎𝑐, and if 𝑏 squared is bigger than four 𝑎𝑐, then you know that there’re gonna be two roots. If 𝑏 squared minus four 𝑎𝑐 equals zero, that square root is gonna be zero. So we’re just gonna have one root.
And maybe a quicker way of spotting that is, if that the 𝑏 squared value is equal to four 𝑎𝑐, means the same thing; that’s just gonna be one root. And if 𝑏 squared minus four 𝑎𝑐 is less than zero, you’re gonna be trying to calculate the square root of a negative number. It’s not gonna work; there’ll be no roots. And that happens when the square of 𝑏 is less than four times 𝑎 times 𝑐.
Okay then. Before we go, I just want you to do these three questions. How many roots do these quadratics have? So we’re gonna be putting the 𝑦-coordinate equal to zero and seeing how many solutions we get. And I want you to do this by analyzing the discriminant in each case. So I’m just- start- I’ve- a pause now and I’m just gonna wait a couple of seconds and then I’m gonna explain the answers.
Right. So in each case, the first thing to do is write down the value of 𝑎, 𝑏 and 𝑐. And then we can use those values to evaluate the discriminant. And the discriminant, remember, is 𝑏 squared minus four 𝑎𝑐.
So with this first question, that’s five squared minus four times two times five. So that’s twenty-five minus forty, which is negative fifteen. So in this case, the discriminant 𝑏 squared minus four 𝑎𝑐 was less than zero and that means that there are no real roots.
Moving on to number two, we can see that 𝑎 is two, 𝑏 is negative four, and 𝑐 is two. So the discriminant is 𝑏 squared minus four 𝑎𝑐. So that’s negative four squared minus four times two times two. Well four squared is sixteen and four times two is eight times two is sixteen. So we’ve got sixteen minus sixteen, so that’s equal to zero. So 𝑏 squared minus four 𝑎𝑐, the discriminant, is equal to zero and that means we’ve got one repeated root.
And for the last question, we’ve got 𝑎 is equal to two, 𝑏 is equal to one cause that means one times 𝑥, and 𝑐 is equal to negative three. And the discriminant 𝑏 squared minus four 𝑎𝑐 is one squared minus four times two times negative three. Now four times two is eight times three is twenty-four. So we’re taking away negative twenty-four, so that means we’re adding twenty-four.
So watch out for these situations; we’re taking away something but because one of those, 𝑐-value in this case, was negative, we’ve got the double negative thing. So the discriminant is twenty-five, which is positive. So that means in the quadratic formula, we’ll be finding the square root of twenty-five, which would be positive or negative five. So we’re adding or subtracting something to our answer there. So the discriminant is greater than zero, so in this case, number three, we’ve got two roots.