Video: Pack 1 • Paper 2 • Question 10

Pack 1 • Paper 2 • Question 10

04:05

Video Transcript

The function 𝑦 equals 𝑓 of 𝑥 is shown on the graph below. Part a) Write down the coordinates of the turning point for 𝑓 of 𝑥. Part b) Estimate the value of 𝑓 of negative 2.4. And part c) Write down the roots of 𝑓 of 𝑥 equals three.

So we’re gonna start with part a. And the keyword here is this term turning point. So what actually is the turning point of a function? Well, there are a couple of properties that are actually gonna help us when we’re looking to find the turning point of our function.

The first is that it’s actually a point where we get a change from an upward or a downward slope to a downward or an upward slope. So if we look at our function, look at our graph, we’ve got a downward slope coming on the left-hand side and an upward slope on the right-hand side. So this would suggest that we’d actually get a turning point at the point that I’ve marked on our graph.

Okay, so let’s look at another property of a turning point just to double check. Another useful property of a turning point that we’re gonna use to actually double check is the fact that a turning point is a maximum or minimum point.

So, for example, I’ve drawn a U-shaped parabola, an inverted U-shaped parabola. And in the U-shaped parabola, you can see it’s a minimum point. And in the inverted U-shaped parabola, you can see that’s a maximum point. So therefore, if we check on our graph, we can see that, great, yeah, this works because actually the point that we marked as our turning point is the minimum point of the graphs, so the minimum point of the function.

Okay, great! So now let’s find the coordinates. So therefore, we can actually see that our 𝑥-coordinate is going to be negative four. And our 𝑦-coordinate is negative three. So therefore, the coordinates of the turning point for 𝑓 of 𝑥 are negative four, negative three. Okay, great! That’s part a solved. Let’s move on to part b.

So in part b, we’re asked to estimate the value of 𝑓 of negative 2.4. So what does this actually mean? What it means is actually where the value of 𝑥 is actually negative 2.4. So what’s the value of our function where 𝑥 is equal to negative 2.4? So in order to actually find this value, what I’ve done is actually marked on where 𝑥 is equal to negative 2.4. And 𝑥 is actually equal to negative 2.4 at this point because each of our small squares are worth 0.2.

Okay, so we’ve got negative 2.4. And then what we do is we go down and see where this actually hits our function. And I’ve drawn a circle to represent this. And then if we read across onto the 𝑦-axis, we can see that the function has a value of negative 0.8. So therefore, we can say that 𝑓 of negative 2.4 is equal to negative 0.8, so great part b solved.

Now let’s move to the final part, part c. So for part c, what we’re trying to do is find the roots of 𝑓 of 𝑥 is equal to three. So again, what does this mean? Well, this means what are the roots where our function is equal to three. So now the first stage was to actually draw on the line 𝑦 equals three because this is what we’re looking for. We’re looking for the points where our function is actually equal to three.

So now what we’re actually looking for are the points where 𝑦 equals three actually intersects out graph, so where they actually meet. We can see that I’ve actually denoted these using a circle. So now, in order to actually find out what the roots are, we actually look down from these points to the 𝑥-axis to actually see what these values are, so the 𝑥 values. When we do, we can see that we have values of negative 6.6 and negative 1.4.

So therefore, we can say that the roots of 𝑓 of 𝑥 is equal to three are negative 6.6 and negative 1.4. And we say this because actually both of these values return a value for our function of three.

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