### Video Transcript

By drawing a graph of the function
๐ of ๐ฅ equals two ๐ฅ squared minus three ๐ฅ, find the solution set of ๐ of ๐ฅ
equals zero.

Our equation is ๐ of ๐ฅ equals two
๐ฅ squared minus three ๐ฅ. Before we start drawing our graph,
itโs 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 ๐ฅ-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 ๐ฅ-values,
represent the domain of the function, what we can plug in for ๐ฅ. And the bottom values represent the
range, what weโll need for the ๐ฆ-values. In these values, we have a range
thatโs lowest point is negative one and highest point is 14. Now, these are just from the points
weโve chosen. That doesnโt mean itโs the range of
the full function. But it does give us an idea of what
the scale of the ๐ฅ- and ๐ฆ-axis should be. We can graph a point at negative
two, 14; negative one, five; zero, zero; one, negative one; and two, two.

When weโre looking at this, we see
that it might be helpful to have an additional point on the ๐ฅ-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 wonโt be perfect, but you can
try to get a smooth curve between the points. And when we do that, since weโre
looking for the solution set of ๐ of ๐ฅ equals zero, weโre looking for the
๐ฅ-intercepts. Weโre looking for the places where
this function crosses the ๐ฅ-axis. The first one is very clear. Itโs the point zero, zero. And the second intersection happens
halfway between one and two. We have a solution when ๐ฅ equals
zero and a solution when ๐ฅ 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 ๐ฅ-values
which when inputted into the function, the output is zero.