The red graph in the figure represents the equation 𝑦 equals 𝑓 of 𝑥 and the green graph represents the equation 𝑦 equals 𝑔 of 𝑥. Express 𝑔 of 𝑥 as a transformation of 𝑓 of 𝑥.
Just so we know what we’re dealing with, let’s begin by labelling our graphs 𝑦 equals 𝑓 of 𝑥 and 𝑦 equals 𝑔 of 𝑥. We’re told that the graph 𝑦 equals 𝑓 of 𝑥 is mapped by some transformation onto the graph 𝑦 equals 𝑔 of 𝑥. So, what is that transformation? Well, when it comes to functions, there are just three that we can perform. These are reflections, enlargements, and translations.
So, which of these three has occurred? We can see that the graph has not been reflected in either the 𝑦- or the 𝑥-axis, nor has it been shifted up or down or left or right. In fact, we can see quite clearly that it has been stretched in the 𝑥-direction. So, we have an enlargement.
And so, we recall two options. If 𝑦 of 𝑓 of 𝑥 is our original function, then the graph of 𝑦 equals 𝑎 times 𝑓 of 𝑥 is an enlargement by a scale factor of 𝑎, that’s s f, in the 𝑦-direction. And the graph of 𝑦 equals 𝑓 of 𝑏 times 𝑥 is an enlargement by a scale factor of one over 𝑏 in the 𝑥-direction. Now, it’s important to realise that 𝑎 and 𝑏 are real constants. So, what has happened to our graph of 𝑦 equals 𝑓 of 𝑥? Well, we can see it has been stretched outwards, that’s horizontally. So, it’s been stretched in the 𝑥-direction.
Now, let’s pick a couple of points on the curve, say the point with coordinate one, zero. This has been stretched and maps onto the point with coordinates two, zero. Similarly, the points with coordinates negative two, zero has been stretched and maps onto the point with coordinates negative four, zero. And so, we can see that the transformation that maps 𝑦 equals 𝑓 of 𝑥 onto 𝑦 equals 𝑔 of 𝑥 is a horizontal stretch or enlargement by a scale factor of two.
Now, if we go back to the definition, we notice that 𝑓 of 𝑏 of 𝑥 was an enlargement by a scale factor of one over 𝑏. That’s the reciprocal. And this means the value of 𝑏 we need that gives us a scale factor of two is one-half. And so, 𝑔 of 𝑥 equals 𝑓 of one-half 𝑥 or 𝑓 of 𝑥 over two.