here we're asked to graph
a function the first thing i notice is
that this function has a degree of 2.
it's a quadratic function and therefore
the graph is going to be a parabola
notice here that the leading coefficient
a is negative
and so that tells me this parabola will
open down
next i want to determine the y-intercept
now the y-intercept
occurs when x is zero
so by inspection we can see that when x
is zero
the y value will be negative nine
so zero comma negative nine is the
y-intercept
to find the x-intercept remember we set
y equal to zero in other words g of x
equal to zero in this case
and then solve so we have zero equals
negative 4x squared plus 12x
minus 9. so here we have a quadratic
equation equal to zero we need to solve
this
let's go ahead and multiply both sides
by negative one
in that case we get 0 equals
4x squared minus 12x
plus 9. and then this
quadratic expression here factors
factoring 4x squared as 2x times 2x
and 9 times 3 times 3 we can see our
inner product is 6x
and our outer product is 6x adds up to
negative 12x if they're both
negative so we can write then 2x
minus 3 equals 0 or
2x minus 3 equals 0. we're going to get
the same
value for both both of these linear
equations here we have x equals
3 halves and or x equals three-half so
in other words we have a double root
that tells me there's only one
x-intercept
and that is at one-and-a-half or
three-halves comma
zero once we determine the intercepts we
then want to
find the vertex
we we could find the x value of the
vertex using the formula
x equals negative b over 2a
now when you use this formula remember
you want to use your original
function here so we have negative b
or negative 12 over 2 times
a 2 times negative 4
which is negative 12 over negative 8
reduces to 3 halves
now to find the corresponding y value we
need to determine
g of 3 halves
now we can do that by substituting three
halves here for each instance
of x
and so doing that we can see here we
have
negative four times three halves squared
which is nine
fourths plus
uh two divides into 12 six times and
then six times three is 18
minus the nine these fours cancel so you
have negative 9
plus 18 minus 9 and certainly that
equals 0.
now we knew that would be the case
because the x intercept had an x value
of 3 halves as well
that tells me the vertex is the same
here
as the x-intercept three-halves comma
zero
okay with the y-intercept the
x-intercept and the vertex
we're ready to graph this parabola
okay so we have a y-intercept at zero
comma negative nine
and x-intercept at three-halves comma
zero three halves is
the same as one and a half so one and a
half comma zero is also
the vertex now we should label that
and in this case we only have two points
and that doesn't seem like enough
to actually graph a parabola so we need
at least one more point
let's see what happens when x is three
okay when x is three then we need to
find the corresponding y value in other
words we need to calculate
g of three so in this case we have
negative four
times three squared right plus twelve
times three minus nine
okay so negative four times 9 plus
36 minus 9 or
negative 36 plus 36 minus 9
and that gives us a result of negative
9. so that tells me the extra point we
can use here
is 3 comma negative 9.
so 3 comma negative 9 gives us that
third point
and then we can sketch the graph of this
parabola
opening downward
and so here we have a rough sketch of
the graph of g
