Video Transcript
If 𝑓 of 𝑥 is equal to 𝑥 squared
minus 𝑥 minus six and the solutions to the equation 𝑥 squared minus 𝑥 minus six
equals zero are three and negative two, then what are the points where 𝑓 of 𝑥
intercepts the 𝑥-axis?
In this question, we have been
given a quadratic function 𝑓 of 𝑥 equals 𝑥 squared minus 𝑥 minus six. We have also been told that the
equation 𝑥 squared minus 𝑥 minus six equals zero has two solutions, which are
three and negative two.
We can recall that the solution set
of a quadratic function can be found by setting 𝑓 of 𝑥 equal to zero. In our case, we have that when 𝑓
of 𝑥 is equal to zero, 𝑥 squared minus 𝑥 minus six is also equal to zero. Hence, we can say that the
solutions to 𝑓 of 𝑥 are three and negative two. Finding this is very useful in
helping us solve the question, since we know that the points where 𝑓 of 𝑥 will
intercept the 𝑥-axis will be when 𝑓 of 𝑥 is equal to zero.
Let’s take a minute to think as to
why this is the case. The equation we are sketching the
graph of is 𝑦 equals 𝑓 of 𝑥. The 𝑥-axis is the same as the
equation 𝑦 equals zero. Hence, the points where 𝑓 of 𝑥
intercepts the 𝑥-axis will be when these two lines intersect. So that is when 𝑓 of 𝑥 is equal
to zero.
We have already found that when 𝑓
of 𝑥 equals zero, 𝑥 is equal to three or negative two. Hence, these are the 𝑥-coordinates
of the points where 𝑓 of 𝑥 intercepts the 𝑥-axis. Since both of these points are on
the 𝑥-axis, we know that their 𝑦-coordinates must be zero.
Therefore, our solution to this
question is that the points where 𝑓 of 𝑥 intercepts the 𝑥-axis are three, zero
and negative two, zero. Here is a quick sketch of what a
graph of 𝑓 of 𝑥 may look like. We can see that it intersect the
𝑥-axis at three, zero and negative two, zero.
