Split each sentence from the text:  10 metres is probably the length of the race. Now, you could have combined that with your previous answer and said that actually Child 2 didn't run 10 metres. They actually only ran 8 metres. But to get 10 metres away from where they should have started, it took them 8 seconds. And the last one, there's some different answers you could have come up with here. You might have said something like, they don't run at a steady pace. Normally you'd accelerate at the beginning and then slow down later on, whereas a straight line graph suggests you're at the same speed all the way through. For a plumber, 0.80 suggests that a job that takes zero hours costs £80. You might have said that this was a call out fee. For B, you should have £120. For C, 10 hours. And then the price per hour is actually the gradient of this line. If you think about every hour you add on, how much does the price go up? We can see from the line that if you add on two hours, the price goes up £20. So for every hour, it's £10. So £10 per hour. Well done. We're now going to have a go at reviewing our quadratic graphs. So we can use a table of values to plot quadratic graphs too. Let's have a look at y equals x squared. So if we square our x values, remembering that when we square a negative value, we get a positive value, and then we can plot our values on a graph. Of course, this is not a linear equation, so it does not form a linear graph. Therefore, we cannot draw a straight line. We can see that really clearly. y equals x squared is a quadratic equation. Its graph is called a parabola. And what we can do is draw a curve instead. Jacob has noticed that we've got a line of symmetry, and in this case it's the y-axis. Okay, let's have a go with some trickier quadratic equations. We're going to do y equals 2x squared minus 4x. But we do exactly the same thing. We just need to substitute the required values for x. I would suggest that you start with zero and then the positive values and then go back to the negative values. The positive values are generally easier. And then you can look at some patterns to make sure you get the right values for the negative x. If you're using a calculator, make sure that you're putting brackets around your substituted value. Right, we've done the first three. Sam says, have we done something wrong? The y-coordinate is zero again. What do you think? y-coordinate is zero again. What do you think? No, our working out is perfect. It's just that there is a line of symmetry somewhere on a parabola. Therefore, it is common for there to be repeated values in the table. It doesn't always happen, but it is usual with a quadratic graph to see repeated values in the table. Let's finish off our positive values. Let's have a think now. What do we expect x equals negative two and x equals negative one to produce? See if you use Jacob's idea about symmetry. OK, let's try it out. We should have 6 and 16. I wonder if that's what you predicted looking at those values. And you can see we have got a line of symmetry because the values are repeated either side of when x is 1. Alex is having a go at plotting this one. is 1. Alex is having a go at plotting this one. I think when x is negative 1, the y value will be 4 because parabolas have a line of symmetry. I'd like you to substitute into this equation to see if Alex is correct. Off you go. Right, whether you're doing this by hand or using a calculator, you need to make sure you put brackets around the value you're substituting. So we should have negative one all squared plus two lots of negative one plus one. Negative one squared is one, so we've got one, add negative two, add one, and that gives us zero. So Alex is incorrect. There is going to be a line of symmetry somewhere, but it's not always the y-axis. So there could be a line of symmetry somewhere else in the table, or there might not even be in the table of values. It might be somewhere else on the graph. So like with linear equations, we can solve quadratic equations using their graph. So let's do x squared plus 4x plus 1 equals 6. First, we can draw the graph. And then we want to know when this is equal to 6. So if you draw the line y equals 6, you'll see that we have two solutions. Often when you have quadratic equations, there are two solutions, but not always. Our two solutions in this case are x equals negative 5 and x equals 1. Like we just said, there could be two solutions or there could be one solution or no solutions. Let's have a look at some examples. So here is the graph of y equals 2x squared minus 4x plus 1. If I want to know when that equals 7, you'll see we have two solutions. When that equals 3, again we have two solutions. We might not be able to tell exactly what they are because they're between integer coordinates. But we do have two solutions and they're approximately negative 0.4 and 2.4. So where on the graph is that only one solution? Can you write an equation that has one solution? See if you can do this. can do this. Well, if you spotted it's at that very bottom point of the curve, it's where x is 1 and y is negative 1. What that means is the equation 2x squared minus 4x plus 1 equals negative 1 only has one solution. Can you think of an equation then that will have no solutions? Okay, it's going to be anywhere below that bottom point of the curve. So for example, 2x squared minus 4x plus 1 equals negative 2. Or essentially, that expression equal to any value less than negative 1. And it's that point of the graph underneath that bottom point of the curve. We can add a context, just like we did with linear graphs. So this is the horizontal and vertical distance of a shot put throw. Why do you think the graph stops when we get to 20, 0? OK, well, that's going to be where the shot put lands on the ground again, because it can't go lower than 0 metres above the ground, unless it's going through the ground, of course. If we want to know the horizontal distance when the shot put is a height of three, we find a height of three. And then we read down to find our horizontal distances. The challenge with these questions is always reading the context and understanding what the graph is showing you. What is the highest point that the shot put reaches? Can you find it? So the highest point is four metres, making sure you're reading off the right value. Quick check then. So this parabola models an athlete completing a pole vault jump. You've got horizontal and vertical distance. What was the highest point reached by the athlete? Off you go. out of those options it has to be 6.25 right how far has the athlete traveled horizontally when they are four meters above the ground what do you think okay so we need to find four meters above the ground. And you'll see that there's two solutions, one when the horizontal distance is 0.2, then obviously on their way back down again, when the horizontal distance is 0.8. Time for you to have a go. I'd like you to use the table of values to plot the parabola with equation x squared minus 2x minus 3 and then use it to find the solution to those equations. Off you go. For question two, I would like you to read the context carefully and then use that to answer the questions, making sure you're thinking about units and full sentences where necessary. Off you go. And this time we've got an Olympic diver. Again, make sure you read the graph carefully and then answer the questions. Off you go. Let's have a look at our answers then. I've used graphing software, so my graph is perfect. However, if you plotted this by hand, it might not be exactly the same as mine. What you want to do is check for key coordinates. Does it go through the coordinate negative 2, 5? Does it go through the coordinate 0, negative 3? Does it have a lowest value at 1, negative 4? And does it have that line of symmetry? Use the table of values to check you've got the right points. Your solutions then, roughly negative 1.8 and 3.8. Again, that's going to depend a little bit on how your graph is drawn. You should get exactly negative 1 and 3. And you can see that from the table of values as well. You can see we've got negative 1, 0 and 3, 0. And you've got one solution of x equals 1 for that last one. And again, you can see that in your table of values. For our penguin, then, can you make sure you've checked that your table of values is correct? And again, you should have a nice, smooth curve for your parabola. You should have a line of symmetry when x equals 4. Why did the table of values start at zero? Well that means zero minutes, so that's when the penguin is starting the dive. It does not make sense to have negative values for minutes. The deepest point that the penguin got to was 160 It was underwater for eight minutes and then 70 meters below sea level was after one minute. And then they returned again after seven minutes. Penguins can actually dive under the water for a really long time. And some emperor penguins can actually get to a depth of 300 meters. emperor penguins can actually get to a depth of 300 meters. So our Olympic diver, well done if you spotted that the person writing that article has not understood what the graph is about. The x-axis is representing horizontal distance, not time. There's no way to gather any information about time from this graph. It's not uncommon for graphs to be incorrect or for people to misinterpret graphs. So it is something that's actually worth keeping your eye out for when you're watching other people working with graphs. Something like 11 meters was the highest point the diver reached and they had moved 0.2 meters horizontally to reach this height. If you think about diving off a diving board, you're obviously going to have to go forwards as well to clear the diving board before you start falling back towards the water. See, 0.410 is the moment where the diver was in line with the starting point again. So like we said, they need to have cleared the diving board so it has moved 0.4 meters horizontally and is back at the same height as where they jumped from. Well I hope you feel like you're really secure now in your understanding of graphs and that you're going to be able to use those going forward. We've looked at plotting linear graphs and the different ways that you might want to go about doing that including a table of values and this idea of gradient and y-intercept. We can use linear graphs to find solutions to equations. And we can put those in context as well and find solutions to problems. We can do exactly the same with linear graphs as with quadratic graphs. Just remembering that they're not going to form a straight line, so you have to be able to draw a curve as neatly as you can through the points, or use graphing software and technology to get it perfect. Remember as well that we can have no solutions, one solution or two solutions from a quadratic equation, and you've seen how we can tell that from the shape of the graph. And then we had some fun playing around with some different real life scenarios, some different Olympic sports, and we found out some fun stuff about penguins. Thank you for joining me in today's lesson and I look forward to seeing you again. again.