Video: AQA GCSE Mathematics Higher Tier Pack 5 β€’ Paper 2 β€’ Question 23

Here is the graph of 𝑦 = 𝑓(π‘₯), where 𝑓(π‘₯) is a quadratic function. Write down the solutions of 𝑓(π‘₯) = 3.

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Video Transcript

Here is the graph of 𝑦 is equal to 𝑓 of π‘₯, where 𝑓 of π‘₯ is a quadratic function. Write down the solutions of 𝑓 of π‘₯ is equal to three.

This might look a little tricky. But let’s just see what’s happened. At the start, we’re told the graph is for the equation 𝑦 is equal to 𝑓 of π‘₯. This just means some function of π‘₯. And in fact, we’re told it’s a quadratic function and we could have spotted that because that graph has the smiley face shape. Then, we’re told to find the solution of 𝑓 of π‘₯ is equal to three.

Notice how the 𝑦 in the first equation has been replaced with three in the second. This means we need to find the solutions of our equation when 𝑦 is equal to three. And we can draw the line 𝑦 is equal to three to help us find these. Remember the line 𝑦 equals three is the horizontal line that passes through the 𝑦-axis at three, as shown.

Notice that the line 𝑦 equals three intersects the curve at two places. The π‘₯-values for these points of intersection will be the solutions to our equation. Let’s add some lines down to our π‘₯-axis from these points. To find the solutions, we need to read the values of π‘₯ from our graph. Before we do though, let’s check the scale.

We can see that five small squares on our π‘₯-axis represent one. We divide by five to work out the value of one small square and that’s one-fifth, which is equal to 0.2. This means the positive solution for π‘₯ is 3.6. It’s three little squares above the number three. And the negative solution for π‘₯ is one little square below the number negative two. So it’s negative 2.2.

The values of π‘₯ when 𝑦 is equal to three and therefore the solutions for 𝑓 of π‘₯ is equal to three are π‘₯ is equal to negative 2.2 and π‘₯ is equal to 3.6.

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