Video: AQA GCSE Mathematics Foundation Tier Pack 3 • Paper 2 • Question 25

(a) Complete the table of values for the function 𝑦 = 3 − 𝑥 − 𝑥². (b) Draw the graph of 𝑦 = 3 − 𝑥 − 𝑥² for 𝑥-values between −2 and 2.

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Video Transcript

Part a) Complete the table of values for the function 𝑦 equals three minus 𝑥 minus 𝑥 squared. Part b) Draw the graph of 𝑦 equals three minus 𝑥 minus 𝑥 squared for 𝑥 values between negative two and two.

So for part a, the first thing we need to do is work out our 𝑦 values given the certain 𝑥 values that we have. So we’re gonna to start with 𝑥 is equal to negative two. Well, what we’re gonna do to find out our 𝑦 value is we’re going to substitute in negative two for 𝑥 into every 𝑥 in our function. So if we do that, we’re gonna get three minus then we’ve got negative two minus and then negative two all squared. So we’re gonna get 𝑦 is equal to three plus two. And we get plus two because if you subtract a negative, then it turns into a positive or a plus.

So we’ve got three plus two then minus four. And we’re gonna get that because you got negative two squared. Well, a negative multiplied by a negative is a positive, and two multiplied by two is four. So we get four. So therefore, we’ve got three plus two minus four. So therefore, our first 𝑦 value is going to be equal to one. And that’s because three add two is five, take away four is one.

Okay, great, now let’s move on to the next value. Well, next, we have 𝑥 is equal to zero. So therefore, we’re gonna substitute this in for our 𝑥 values in our function. So therefore, this one is gonna be quite straightforward because we’re gonna have 𝑦 is equal to three minus zero minus zero squared, which gonna give us a result of three. So therefore, we can say that 𝑦 is equal to three. So great, this is another 𝑦 value. And this is the 𝑦 value when 𝑥 is equal to zero.

And then next, what we do is we substitute in 𝑥 is equal to one. So 𝑦 is equal to three minus one minus one squared. So therefore, we’re gonna get 𝑦 is equal to one. And that’s cause three minus one is two. And then minus one squared, well one squared is just one. So it’s minus another one, which leaves us with 𝑦 is equal to one.

So then, finally, we substitute in 𝑥 is equal to two. So we have 𝑦 is equal to three minus two minus two squared. So we’re gonna get three minus two minus four, which is gonna give us a value of 𝑦 of negative three. So therefore, we could say when 𝑥 equal to two, 𝑦 is equal to negative three. So therefore, we’ve solved part a, because we completed the table of values for the function 𝑦 equals three minus 𝑥 minus 𝑥 squared. And we’ve done that here.

And it’s worth noting that we can check to make sure it looks right. And because we’re dealing with a quadratic — we’ve got an 𝑥 squared term — then we’re looking for a symmetrical shape. And we can see that we’ve got one, three, three, one. So we can see that that actually produces some symmetrical 𝑦 values when you have our 𝑥 inputs. So we’ll also gonna get a look at that when we’re gonna look at the graph that we’re going to draw for our function.

Okay, so now let’s move on to part b and draw that graph. Well, now before we draw the graph and plot the values that we’ve got from the table, let’s think about what it should look like. Well, we’ve got 𝑦 equals three minus 𝑥 minus 𝑥 squared. So we have an 𝑥 squared term. So therefore, we know that this is gonna be a quadratic graph. So we’re gonna have a symmetrical parabola as our shape. And this parabola can take one of two forms. It can be a U-shaped parabola or an inverted U- or n-shaped parabola. But how are we going to decide which one that our function is going to have?

Well, it’s all down to the sign, the sign of our 𝑥 squared term. So in our case, we have a negative 𝑥 squared. And we know that if we have a negative 𝑥 squared term, then our function graph is going to be an inverted or n-shaped parabola. But if we had a positive 𝑥 squared term, then we’d have a U-shaped parabola.

So now we know what the shape is going to look like. Let’s plot the points to see if it takes this form. So we’re gonna plot the first point. And we’ve got the 𝑥-coordinate is negative two from our table. And we calculated the 𝑦-coordinate to be one. So therefore, I’ve put a cross at that point. So it’s negative two on the 𝑥- or horizontal axis and one on the vertical axis, which is our 𝑦-axis.

And then our next point is negative one, three, so negative one on the 𝑥-axis, three on the 𝑦-axis. Then we have a zero, three, so again zero on the 𝑥-axis, three on the 𝑦-axis. Then we have one, one, because that’s our next point. 𝑥 equals one and 𝑦 equals one, so one on the 𝑥-axis and one on the 𝑦-axis. And then our final point is two, negative three, so two on the 𝑥-axis and negative three on the 𝑦-axis. So great, we’ve got our points plotted.

So now let’s draw our curve. Well, we’ve already established that it’s an n-shaped or inverted U-shaped parabola. We can see that our points are gonna have that general shape. So now we’ve got to draw one of these. So now we’ve connected our points and drawn ourselves a smooth parabola. So as we said, this one is an inverted U-shape or an n-shaped parabola.

So there we have it. We’ve solved part a and part b because we’ve drawn the table of values and then we’ve constructed the graph of the function 𝑦 equals three minus 𝑥 minus 𝑥 squared.

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