here we're asked to graph x equals y
squared minus y
minus six now the first thing i notice
is that this is quadratic in y
that tells us that the graph will be a
sideways parabola
now since the leading coefficient is 1
that's positive
we know that this sideways parabola is
going to open to the right
let's begin by finding the x intercept
now to find the x intercept we set y
equal to zero
so in this case we have x equals
zero squared minus zero minus six
in other words x equals negative six
and so that tells me the x intercept
will be negative six comma
zero next let's find the y
intercepts and to find the y
intercepts we set x equal to zero
so in this case we have zero equals
y squared minus y minus six
which is quadratic and it does
factor if we write y and y
and 2 and 3
we'll we'll have a negative 1y if the 3
is negative here and the 2 is positive
and then set each factor equal to zero
and that gives us two answers y equals
negative two
or y equals three
and so that tells us we have two
y-intercepts one
at zero comma negative two
and another at zero comma three
now finally we wanna find the
vertex
and to do that here we will use vertex
form recall vertex form is
x equals a times the quantity y minus k
squared plus h
if we can write the original equation in
that
form then the vertex will simply be h
comma k
now to do that we'll begin by making
room
for the constant that completes the
square
so we'll be adding some number and then
at the same time subtracting the same
exact number
now the number that completes the square
remember
is b over 2
squared so in this case b is
negative 1. so negative 1 over 2
squared gives us 1
4. and so that's the number we're going
to use to complete the square
we'll add 1 4 and subtract 1 4
to the right side of the equation and
then
factor this resulting trinomial here
and in this case we get
well we should always get a perfect
square so we have y times y
1 one-fourth factors as one-half times
one-half
and adds up to negative one y
here if they're both negative
now negative 1 4 minus 6
adds up to negative 25
4. and so we have x
equals y minus one half
squared minus 25
fourths and we have vertex form
in this form we can read the vertex
let me take an extra step to be
absolutely clear
y minus one-half one-half will be the
k squared plus
negative 25 fourths
okay so here we can see then that the
vertex
the h comma k
in this case is negative
25 4 comma
one half
okay once we have our x and y intercepts
and our vertex
we're ready to graph this
okay so remember we found the
x-intercept to be at
negative six comma zero
and then we found the y-intercepts one
at
excuse me let me fix this zero
comma negative two and
zero comma three right
those reversed
okay so zero comma negative two is here
and 0 comma 3
is there now the vertex we found
was negative 25 4 or negative
6 and a quarter comma one half
so negative six and a quarter is about
right there
let's go ahead and label that and then
finally we can see that there'll be a
point of symmetry here at
negative six comma
one okay so now we're ready to
sketch this sideways parabola
and so here we have a rough sketch of
x equals y squared minus y
minus 6.
