Video Transcript
The figure shows the graph of 𝑦
equals 𝑓 of 𝑥. Which of the following is the graph
of 𝑦 equals a half 𝑓 of 𝑥?
Let’s begin by looking at the
equation of the transformed function. When we multiply by a scalar, that
is, a real constant, that represents a dilation or enlargement of some
description. In fact, when we multiply the
entire function 𝑓 of 𝑥 by some scalar, we get a vertical dilation or enlargement
by a scale factor of that number. And so, here we’re going to stretch
the original graph vertically by a scale factor of one-half. That’s going to look like a
vertical compression. To identify the correct graph,
we’ll identify some of the key points on our graph.
Firstly, let’s consider this point
here. It passes through the 𝑦-axis at
two. When we compress our graph or
stretch it vertically by a scale factor of one-half, this will now pass through a
value of 𝑦 half the size. It’s going to pass through the
𝑦-axis at zero, one. Similarly, let’s take the point at
1.5, negative 0.6. We’re going to halve the value of
the 𝑦-coordinate. The 𝑥-coordinate still remains
unchanged, so it’s going to be 1.5, negative 0.3. And so, it’s going to look a little
something like this. If we compare this to the graphs
we’ve been given, we see that the only one that matches this criteria and the only
one in fact that passes through the 𝑦-axis at one is (B). So, (B) is the graph of 𝑦 equals a
half 𝑓 of 𝑥.
Let’s see if we can identify the
equations of the other graphs. Looking at graph (A), we can see
it’s actually been stretched by a scale factor of two. And so, the equation of this one
must be 𝑦 equals two times 𝑓 of 𝑥. Graph (C), however, has been
compressed by a scale factor of a half. But this time, that’s in the
horizontal direction. To achieve a horizontal dilation by
a scale factor of one-half, we need to multiply the values of 𝑥 by two. So, the equation of this graph is
𝑦 equals 𝑓 of two 𝑥.
Then, if we look at graph (D), we
see something similar has occurred. This time it’s stretched in a
horizontal direction but by a scale factor of two. To achieve this, we need to
multiply all the values of 𝑥 by one-half. So, graph (D) is 𝑦 equals 𝑓 of a
half 𝑥. And graph (E) is a different beast
altogether. This represents a combination of
stretches. It stretched vertically by a scale
factor of two and horizontally by a scale factor of two. And so, its equation is, in fact, a
combination of (A) and (D). It’s 𝑦 equals two times 𝑓 of a
half 𝑥. The correct answer here, though, is
(B).
