Video Transcript
True or False: If the point
negative two, negative five is the vertex of the graph of the quadratic function 𝑓
of 𝑥 is equal to 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, where 𝑎 is a negative number,
then the solution set of the equation 𝑓 of 𝑥 is equal to zero is the empty
set.
In this question, we’re given some
information about a quadratic function 𝑓 of 𝑥. We’re told that the value of 𝑎,
its leading coefficient, is a negative number. We’re also told that the point with
coordinates negative two, negative five is the vertex of the graph of this quadratic
function. We need to use this information to
determine if the solution set of the equation 𝑓 of 𝑥 is equal to zero is the empty
set.
And to do this, let’s start by
recalling what we mean by the solution set of an equation. It’s the set of all solutions to
the equation. So, for the equation 𝑓 of 𝑥 is
equal to zero, it’s the set of all values of 𝑥 such that 𝑓 evaluated at 𝑥 is
zero. And we need to determine if it’s
true whether the solution set to this equation is the empty set, which means it has
no solutions.
There’s several different ways we
could go about doing this. For example, we could try and find
the solutions algebraically. However, this is quite
difficult. So, instead, because we’re given
the sign of the leading coefficient of our graph and the coordinates of its vertex,
we’re going to do this graphically. Let’s start by sketching a graph of
the function. And to do this, we can recall all
quadratic curves have a parabolic shape. And in particular, there’s two
orientations for a parabola which are determined by the sign of the leading
coefficient.
If the leading coefficient is
negative, then we say that the parabola opens downwards. However, if the leading coefficient
is positive, then we say that the parabola opens upwards. And in our case, we’re told that
the sign of the leading coefficient 𝑎 is negative. So the shape of our parabola will
open downwards. We can also recall that the turning
point of these parabolas is called the vertex of the parabola. If the parabola opens downwards,
then the 𝑦-coordinate of the vertex tells us the maximum output of the
function. And if the parabola opens upwards,
then the 𝑦-coordinate of the vertex tells us the minimum output of the
function. And it’s also worth noting all
parabolas are symmetric through the vertical line through their vertex.
And now we can sketch the graph of
our parabola. We’ll start by drawing a pair of
coordinate axes and marking the coordinates of the vertex negative two, negative
five. We then want to sketch a parabola
with this point as the vertex which opens downwards. For example, we might get the
following. However, it’s worth noting we don’t
know how narrow or wide this parabola will be. For example, we might have a wider
parabola which still has the point with coordinates negative two, negative five as
its vertex and opens downwards. And we might have a more narrow
parabola, such as the following. We don’t know the exact shape of
this curve.
However, we can notice something
interesting all of the parabolas have in common. They all have the vertex negative
two, negative five. And now since our parabola opens
downwards, we know the 𝑦-coordinate of the vertex tells us the maximum output of
the function. Its maximum output is negative
five, and this occurs when 𝑥 is equal to negative two. But if the maximum output of the
function is negative five, the function will never output zero. So it has no solutions. So the equation has no
solutions. So its solution set is the empty
set.
And it’s worth noting this is
equivalent to saying that the graph of the function has no 𝑥-intercepts because an
𝑥-intercept would be the values of 𝑥 where the function outputs a value of
zero. So we could’ve equivalently
determined that the statement is true by noting none of our sketches will pass
through the 𝑥-axis. In either case, we were able to
show if the point negative two, negative five is the vertex of the graph of a
quadratic function with negative leading coefficient, then it’s true the solution
set of the equation 𝑓 of 𝑥 is equal to zero must be the empty set.
