The function 𝑦 equals 𝑓 of 𝑥 is
shown on the graph below. Part a) Write down the coordinates
of the turning point for 𝑓 of 𝑥. Part b) Estimate the value of 𝑓 of
negative 2.4. And part c) Write down the roots of
𝑓 of 𝑥 equals three.
So we’re gonna start with part
a. And the keyword here is this term
turning point. So what actually is the turning
point of a function? Well, there are a couple of
properties that are actually gonna help us when we’re looking to find the turning
point of our function.
The first is that it’s actually a
point where we get a change from an upward or a downward slope to a downward or an
upward slope. So if we look at our function, look
at our graph, we’ve got a downward slope coming on the left-hand side and an upward
slope on the right-hand side. So this would suggest that we’d
actually get a turning point at the point that I’ve marked on our graph.
Okay, so let’s look at another
property of a turning point just to double check. Another useful property of a
turning point that we’re gonna use to actually double check is the fact that a
turning point is a maximum or minimum point.
So, for example, I’ve drawn a
U-shaped parabola, an inverted U-shaped parabola. And in the U-shaped parabola, you
can see it’s a minimum point. And in the inverted U-shaped
parabola, you can see that’s a maximum point. So therefore, if we check on our
graph, we can see that, great, yeah, this works because actually the point that we
marked as our turning point is the minimum point of the graphs, so the minimum point
of the function.
Okay, great! So now let’s find the
coordinates. So therefore, we can actually see
that our 𝑥-coordinate is going to be negative four. And our 𝑦-coordinate is negative
three. So therefore, the coordinates of
the turning point for 𝑓 of 𝑥 are negative four, negative three. Okay, great! That’s part a solved. Let’s move on to part b.
So in part b, we’re asked to
estimate the value of 𝑓 of negative 2.4. So what does this actually
mean? What it means is actually where the
value of 𝑥 is actually negative 2.4. So what’s the value of our function
where 𝑥 is equal to negative 2.4? So in order to actually find this
value, what I’ve done is actually marked on where 𝑥 is equal to negative 2.4. And 𝑥 is actually equal to
negative 2.4 at this point because each of our small squares are worth 0.2.
Okay, so we’ve got negative
2.4. And then what we do is we go down
and see where this actually hits our function. And I’ve drawn a circle to
represent this. And then if we read across onto the
𝑦-axis, we can see that the function has a value of negative 0.8. So therefore, we can say that 𝑓 of
negative 2.4 is equal to negative 0.8, so great part b solved.
Now let’s move to the final part,
part c. So for part c, what we’re trying to
do is find the roots of 𝑓 of 𝑥 is equal to three. So again, what does this mean? Well, this means what are the roots
where our function is equal to three. So now the first stage was to
actually draw on the line 𝑦 equals three because this is what we’re looking
for. We’re looking for the points where
our function is actually equal to three.
So now what we’re actually looking
for are the points where 𝑦 equals three actually intersects out graph, so where
they actually meet. We can see that I’ve actually
denoted these using a circle. So now, in order to actually find
out what the roots are, we actually look down from these points to the 𝑥-axis to
actually see what these values are, so the 𝑥 values. When we do, we can see that we have
values of negative 6.6 and negative 1.4.
So therefore, we can say that the
roots of 𝑓 of 𝑥 is equal to three are negative 6.6 and negative 1.4. And we say this because actually
both of these values return a value for our function of three.