### Video Transcript

Which of the following graphs
represents the equation π¦ equals negative two π₯ squared plus nine π₯ minus
seven?

Our function π of π₯ equals
negative two π₯ squared plus nine π₯ minus seven is given in the general form ππ₯
squared ππ₯ plus π. But before we do anything with the
function, letβs consider each of these graphs. When we look at the graph of (A),
it opens upward and has a vertex in the fourth quadrant. The graph of (B) opens downward and
has a vertex in the second quadrant. (C) opens downward and has a vertex
in the first quadrant. (D) opens downward and has a vertex
in the first quadrant. And (E) opens upward and has a
vertex in the fourth quadrant.

A good strategy to finding the
graph would be to take our equation and find its vertex. When we have an equation in the
general form, we find the vertex to be negative π over two π. Thatβs the π₯-coordinate. So we take negative nine over two
times negative two, which is positive 2.25. This means the π₯-coordinate of our
vertex will be located at 2.25. If we draw the line π₯ equals 2.25
on all five of our graphs, we see the in (A) that intersects the vertex and in (C)
that intersects the vertex. On that basis, (B), (D), and (E)
are not the graphs weβre looking for.

To find the π¦-coordinate of the
vertex, we could plug in 2.25 into our equation. When we solve for that, we get
3.125. And the vertex for (C) is located
at 3.125. We also notice that our π-value is
negative. And that means our graph must open
downward. Option (A) opens upward and has a
vertex in the wrong place. And so we can say that option (C)
is the correct graph here.