Video: AQA GCSE Mathematics Higher Tier Pack 3 β€’ Paper 1 β€’ Question 23

(a) The graph of 𝑦 = cos(π‘₯) for 0Β° ≀ π‘₯ ≀ 360Β° is shown. On the same grid, sketch the graph of 𝑦 = cos(π‘₯ βˆ’ 90Β°) for 0Β° ≀ π‘₯ ≀ 360Β°. (b) On the grid below, sketch the graph of 𝑦 = βˆ’cos(π‘₯) βˆ’ 1 for 0Β° ≀ π‘₯ ≀ 360Β°.

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Video Transcript

The graph of 𝑦 equals cosine of π‘₯ for zero degrees is less than or equal to π‘₯ which is less than or equal to 360 degrees is shown. On the same grid, sketch the graph of 𝑦 equals cosine of π‘₯ minus 90 degrees for zero is less than or equal to π‘₯ which is less than or equal to 360 degrees.

Notice on the graph, the π‘₯-values are given in degrees and the values along the 𝑦-axis are not being measured in degrees. When using a cosine function, the π‘₯-values represent an angle. So the input is an angle. But the output 𝑦 is not measured in degrees.

If you think about the way we use trigonometry with right-angled triangles, the cosine of an angle is equal to its adjacent side length over the hypotenuse. In this example, the cosine of π‘₯ would be four-fifths. This four-fifths represents a ratio of side lengths, not angles.

Back to the problem in hand, considering these five values for π‘₯, we’ll calculate π‘₯ minus 90 degrees for each of these five values. First, zero degrees minus 90 degrees equals negative 90 degrees. Then, 90 degrees minus 90 degrees equals zero degrees. 180 degrees minus 90 degrees equals 90 degrees. 270 degrees minus 90 degrees equals 180 degrees. 360 degrees minus 90 degrees equals 270 degrees.

In our next step, we want to know what the cosine of π‘₯ minus 90 degrees is. The cosine of negative 90 degrees equals zero. Remember this is not being measured in degrees. The cosine zero degrees equals one, cosine of 90 degrees equals zero, the cosine of 180 degrees equals negative one, and the cosine of 270 degrees equals zero.

And now, we need to be careful. The first row is our π‘₯-value. And the third row β€” this cosine of π‘₯ minus 90 degrees β€” is the output. It’s the 𝑦-value because 𝑦 equals cosine of π‘₯ minus 90 degrees. We use this second row to help us calculate our function. But it is not part of the coordinate.

The π‘₯ and the 𝑦 will create the coordinates of the point. When π‘₯ is zero degrees, 𝑦 equals zero. The next point would be 90 degrees, one, then 180 degrees, zero, 270 degrees, negative one, and then 360 degrees, zero.

This question is asking for a sketch of the graph. To make the sketch, we’ll plot off five of these points on the grid: zero, zero; 90, one; 180, zero; 270, negative one; and 360, zero. From here, we need to connect the points and we see the sketch for the graph 𝑦 equals cosine of π‘₯ minus 90 degrees.

Before we move on, it’s worth noting that there is actually a rule about this kind of transformation. 𝑓 of π‘₯ minus 𝑏 shifts the function 𝑏 units to the right. We are subtracting 90 degrees from our π‘₯-value. And that means our function is being shifted by 90 degrees to the right. Remember our π‘₯-axis, the units are degrees. We are moving the function to the right 90 degrees. If you didn’t remember the rule, that’s fine because you can always solve by plugging in values and graphing them.

Now for part b). On the grid below, sketch the graph of 𝑦 equals negative cosine of π‘₯ minus one for zero degrees is less than or equal to π‘₯ which is less than or equal to 360 degrees.

Again, let’s consider these five angles for π‘₯. First, we’ll find the negative cosine of π‘₯. Cosine of zero degrees is equal to one. And that means the negative cosine of zero degrees equals negative one. The negative cosine of 90 degrees equals zero. The negative cosine of 180 degrees equals one. Negative cosine of 270 degrees equals zero. And negative cosine of 360 degrees equals negative one.

Let’s plug these points on our grid: zero degrees, negative one; 90 degrees, zero; 180 degrees, one; 270 degrees, zero; and 360 degrees, negative one. If this is a graph of 𝑦 equals the negative cosine of π‘₯, then the graph 𝑦 equals negative cosine of π‘₯ minus one will be shifted down one on the 𝑦-axis. The negative cosine of π‘₯ minus one for zero degrees would be negative two.

And on the graph, the point zero, negative one becomes zero, negative two. The negative cosine of 90 degrees minus one is negative one: 90 degrees, negative one. The next point becomes 180 degrees, zero. We’re subtracting one from one. We put 180 degrees, zero on the graph. 270 degrees, zero becomes 270 degrees, negative one. And the last point 360 degrees negative one becomes 360 degrees, negative two. To complete the sketch, we fill in lines between the points. And this is the sketch for 𝑦 equals negative cosine of π‘₯ minus one.

And again, we have a rule for this. If we’re taking a function and subtracting 𝑏, then the function is shifted down 𝑏 units. In our function negative cosine of π‘₯ minus one, the transformation is a shift down one unit from the negative cosine of π‘₯. But again, if you didn’t remember the exact rule, you can always use points to plug in to the function, which will give you an accurate sketch.

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