Video: Pack 1 • Paper 3 • Question 20

Pack 1 • Paper 3 • Question 20

06:03

Video Transcript

The graph below shows part of the curve 𝑦 equals negative 𝑥 squared plus seven over two 𝑥 minus one. Part a) By drawing a straight line on the graph, estimate the solutions to the equation negative 𝑥 squared plus three 𝑥 plus three equals zero. 𝑧 is a point on the curve 𝑦 equals negative 𝑥 squared plus seven over two 𝑥 minus one, where 𝑥 is equal to four. Part b) Write down an estimate for the gradient of the graph at point 𝑧.

So for part a, the first thing that we need to do is actually draw a straight line on the graph which is actually gonna help us to estimate the solutions to the equation negative 𝑥 squared plus three 𝑥 plus three is equal to zero. But what straight line we’re actually going to draw? In order to actually work out which straight line we’re gonna draw, what I want to do first is actually look at each part separately to easily differ between our two functions.

So first of all, we’ve got three 𝑥. So we have to think how we’re gonna make our three 𝑥 out of our seven over two 𝑥 and something else. So what we have is seven over two 𝑥 minus a half 𝑥. And that’s because if you have seven over two 𝑥 minus a half 𝑥, we get six over two 𝑥 and six over two 𝑥 is the same as three 𝑥 cause six over two is three. Okay, great, so we’ve now worked out how to find our three 𝑥. Let’s move on to the next component.

So the next term is actually plus three. And we can make that with the negative one from our original function and then plus four. So therefore, we’re gonna actually use these and rewrite our negative 𝑥 squared plus three 𝑥 plus three equals zero as negative 𝑥 squared plus seven over two 𝑥 minus one minus a half 𝑥 plus four equals zero.

And a key part of this is that we have our original function first, which is negative 𝑥 squared plus seven over two 𝑥 minus one. And then, we have the secondary part which is minus a half 𝑥 plus four equals zero. So what we can now do is actually plus a half 𝑥 and minus four from each side. So what we now have is negative 𝑥 squared plus seven over two 𝑥 minus one is equal to a half 𝑥 minus four.

So therefore, our original function is equal to a half 𝑥 minus four. And this is the straight line that we’re gonna actually draw on our graph to help us find the solutions to our equation. So now, we’re actually gonna draw the straight line on our graph. And the way that we actually do that is we’re picking three values for 𝑥 and then plotting those and then drawing a straight line through that connects them. So we know that the equation of the line is gonna be 𝑦 is equal to a half 𝑥 minus four.

So we know that the first value is 𝑥 is equal to negative three. So therefore, 𝑦 is gonna be equal to negative 11 over two which is the same as negative five and a half because if we have a half of 𝑥, that will be a half of negative three which will be negative one and a half. And then, negative one and a half minus four gives us negative five over a half or negative 11 over two.

So we’ve now plotted that point as shown on the graph. So next, we’ve got 𝑥 is equal to zero. And therefore, we’re gonna get 𝑦 is equal to negative four. And that’s because if we had a half multiplied by zero, that gives us zero and a zero minus four gives us negative four. And also, it’s in the form 𝑦 equals 𝑚𝑥 plus 𝑐. And the negative four is actually our 𝑐, so our 𝑦-intercept. So that also tells us that’s where it would cross the 𝑦-axis.

And then, finally, the last point I chose was 𝑥 is equal to three. And this gives us that 𝑦 is equal to negative five over two. And again, it’s a half multiplied by three this time, which gives us one and a half. And one and a half minus four gives us negative five over two or the same as negative two and a half.

So now, what I’ve actually done is drawing the line and actually marked on its equation. So we’ve got 𝑦 equals a half 𝑥 minus four. So now, we need to actually work out the estimate for the solution to the equation negative 𝑥 squared plus three 𝑥 plus three equals zero. And this is actually gonna be where our two graphs actually meet. So now, what I’ve actually done is actually drawn lines up from this point to the points of intersection of the curve and the line to the 𝑥-axis to actually find our solutions of the equation. So therefore, we can see that 𝑥 is equal to either negative 0.8 or 𝑥 is equal to 3.8. So great, we’ve actually solved part a. So let’s move on to part b.

So we now know that 𝑧 is a point on a curve 𝑦 is equal to negative 𝑥 squared plus seven over two 𝑥 minus one, where 𝑥 is equal to four. So what I’ve actually done is actually marked this point on our curve with a pink cross. So now, what I need to do is actually write down an estimate for the gradient of the graph at this point, so our point 𝑧. So now, what I’ve done is I’ve actually drawn a straight line that’s actually gonna represent the gradient of the curve at this point. And the straight line is actually called tangent, so the tangent to the curve at the point 𝑧.

So now, to actually find the gradient of this line, what I’ve done is actually drawn a small triangle here. It could be between any two points. Well, I’ve chosen these two points because they’re nice and clear. And what we want to actually find is the change in 𝑦 over the change in 𝑥.

So first one, would start with the change in 𝑦. And we can see that the two points we’ve picked we got one at 𝑦 equals six and one at 𝑦 equals two. So therefore, we can see that the change in 𝑦 is gonna be negative four cause it’s gone down four. And in the 𝑥-axis, what it actually does is it goes from two to three. So therefore, we can see that there’s a change of one. So we can actually say that it increases by one.

So therefore, in real terms, what we can say is that when 𝑥 increases by one unit, 𝑦 actually decreases by four units. And we know then that actually our formula for the gradient or 𝑚 is equal to the change in 𝑦 over the change in 𝑥. So therefore, we can say that the gradient or 𝑚 is equal to negative four over one. So therefore, we can actually say that an estimate for the gradient of the graph at point 𝑧 is gonna be equal to negative four.

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