Video Transcript
By drawing a graph of the function
𝑓 of 𝑥 equals two 𝑥 squared minus three 𝑥, find the solution set of 𝑓 of 𝑥
equals zero.
Our equation is 𝑓 of 𝑥 equals two
𝑥 squared minus three 𝑥. Before we start drawing our graph,
it’s helpful to have a few coordinates so that we can sketch the curve. We can use the tables to do
this. If we calculate the 𝑥-values,
negative two, negative one, zero, one, two, that should give us some idea of the
shape of this graph. Our first box would be two times
negative two squared minus three times negative two, which equals 14. Next, we have two times negative
one squared minus three times negative one, which is five. When we plug in zero, the result is
zero. When we plug in one, we get
negative one. When we plug in two, we get
two.
The top values, the 𝑥-values,
represent the domain of the function, what we can plug in for 𝑥. And the bottom values represent the
range, what we’ll need for the 𝑦-values. In these values, we have a range
that’s lowest point is negative one and highest point is 14. Now, these are just from the points
we’ve chosen. That doesn’t mean it’s the range of
the full function. But it does give us an idea of what
the scale of the 𝑥- and 𝑦-axis should be. We can graph a point at negative
two, 14; negative one, five; zero, zero; one, negative one; and two, two.
When we’re looking at this, we see
that it might be helpful to have an additional point on the 𝑥-axis, so we could
solve for three. When we plug three into the
equation, we get nine as the output. And that just gives us one more
point to be able to sketch this graph. It won’t be perfect, but you can
try to get a smooth curve between the points. And when we do that, since we’re
looking for the solution set of 𝑓 of 𝑥 equals zero, we’re looking for the
𝑥-intercepts. We’re looking for the places where
this function crosses the 𝑥-axis. The first one is very clear. It’s the point zero, zero. And the second intersection happens
halfway between one and two. We have a solution when 𝑥 equals
zero and a solution when 𝑥 is one and a half which makes our solution set zero and
three-halves or zero and one and a half.
Remember, this is not a coordinate
point. These are two different 𝑥-values
which when inputted into the function, the output is zero.
