Welcome to a video on graphing quadratic functions in standard form. In a previous video,
we graphed quadratic functions in general form or the form of f of x equals a times x squared plus
b times x plus c. So we should know that the graph of this is a parabola. But if we take general form
and we complete the square on the right side, we'll have standard form that looks like this,
where f of x is equal to a times the quantity x minus h squared plus k.
What's nice about standard form is we can tell a lot about the graph just by looking at the form
of the function. If a is positive or greater than zero, the parabola opens up, and if it's negative
or less than zero, it opens down. Also, the vertex of the parabola occurs at the point h, k,
and the vertical line passing through the vertex is the axis of the parabola or axis of symmetry.
One of the things we have to be very careful about is identifying the correct sign of the x and y
coordinates of the vertex, which are identified by h and k. But it has to be in the form of the
quantity x minus h squared plus k. What that means is if we had x minus h, we'd use a positive value
for h, and if it was x plus h, we'd use a negative value of h. However, for k, if it's plus k,
we use a positive k, and if it's minus k, we're going to use a negative value for k.
So for example, if we have y equals the quantity x minus three squared minus one, the x-coordinate
of the vertex would be positive three, and the y-coordinate would be negative one. So
you can think that the x-coordinate is the opposite of what you see,
but the y-coordinate is exactly the same. So if we have y equals the quantity x plus three
squared plus five, the x-coordinate would be negative three in this case,
but the y-coordinate would be positive five. The real reason why this is negative three is because
it actually has to be written in the form of x minus a negative three squared plus five, but
as you already know, we normally write minus a negative as plus a positive.
So we should be very careful when we're identifying the signs of the x and y
coordinates for the vertex. Let's go ahead and take a look at a graph of a quadratic function
in standard form. What we're going to do here is leave h and k equal to zero and change the
value of a. Notice when a is equal to one, the parabola opens up. As we increase the value of
a, it still opens up, but it is stretched vertically. As we decrease the value of a,
it is stretched horizontally, but as soon as a is negative, the parabola opens downward.
Let's go ahead and put the value of a back to one and change the value of h. Remember,
h is the x-coordinate of the vertex. So as you might expect,
changing the value of h shifts the parabola left or right. Notice that when h is positive
four point seven, the equation is in the form of x minus four point seven. And when h is negative,
the equation is in the form x plus the value because they’re really subtracting a negative.
Let's see what happens when we change k. Remember,
k is the y-coordinate of the vertex. So if we increase k,
it shifts the parabola up. If we decrease k, or if k is negative, the parabola is shifted downward.
Let's go ahead and take a look at a couple examples. Let's say we want to graph this
quadratic function by finding the vertex, axis of symmetry, and the intercepts. Well, we should be
able to tell that the value of a would be one, so that tells us the parabola opens up. Next,
h and k would be the coordinates of the vertex. So the value of h would be positive three, and
the value of k would be negative four, so we have three negative four for the vertex.
Remember, the axis of symmetry is a line passing through the vertex,
so the equation of the axis must be x equals three. Again,
we found all of this useful information just by looking at the form of the function.
Let's go ahead and review how to find the intercepts. Remember,
to find the y-intercept, we're supposed to set x equal to zero and solve for y. So we
would have y equals zero minus three squared minus four. That would be negative three minus
four, which is negative seven. So the y-intercept would be zero negative seven.
And then lastly, in order to find the x-intercepts,
we're supposed to set y equal to zero and solve for x. So we would
have zero equals the quantity x minus three squared minus four. So add four to
both sides and then square root both sides of the equation. Don't forget the plus or minus.
So on the left, we have plus or minus two equals x
minus three. Add three to both sides, and we’d have three plus or minus two.
So we have two x-intercepts. The first one would be three plus two,
which would be five, and the second x-intercept would be three minus two, which would be one.
That should be plenty of information to make a nice graph. Let's go ahead
and sketch it. We'll plot the vertex at three negative four. Let's go ahead
and sketch the axis of symmetry in next. The y-intercept is zero negative seven. Here now,
because we have the axis of symmetry, we know there must be a mirror image of
this point three units to the right of the axis of symmetry here. That'll be helpful.
And then lastly, we have our two x-intercepts, five and one. Again,
that looks good because they're symmetrical across the axis of symmetry. So now we can
just make a nice parabola passing through these five points. Let's go ahead and try another one.
Okay, we notice that it's in standard form. a is equal to negative two, so this parabola opens
downward. The vertex is h, k, so since we have x plus one, the x-coordinate would be negative one,
and we have plus eight on the end, so the y-coordinate would be eight, and therefore,
the equation of the axis must be x equals negative one.
Now, it doesn't ask us to find the x and y intercepts; it just says to find at least two
additional points. I'm going to go ahead and at least find the y-intercept. So to find the
y-intercept, we set x equal to zero and solve for y. We have y equals negative two times the
quantity of zero plus one squared plus eight. That would be one squared plus eight,
which would be negative two plus eight, that would be six. So our y-intercept would be zero six.
Let's go ahead and sketch this information and see what we have. The vertex is
negative one eight, the axis of symmetry is the vertical line passing through that point, and we
have the y-intercept of zero six here. There must be a mirror image of this point one unit to the
left of the axis of symmetry there, and we do have some good information to make a nice graph.
But let's go ahead and find the x-intercepts again just to review. Remember, to find the
x-intercepts, we set y equal to zero. So we’d have zero equals negative two times the quantity of x
plus one squared plus eight. So we're going to subtract eight and then divide
by negative two. That would give us four equals the quantity x plus one squared.
Now we square root both sides of the equation. We have our plus or minus there. Looks like we
have plus or minus two equals x plus one, so we subtract one on both sides, and we have negative
one plus or minus two. So the first x-intercept would be negative one plus two, which would be
one zero, and the second intercept would be negative one minus two, which would be negative three zero.
So we were pretty fortunate here; these came out very nice. Let's go ahead and plot those
and complete our graph. We have one, and notice they are two units to the right of the axis,
and negative three is two units to the left of the axis, which is a good indication that
we're doing this correctly. Let's go ahead and sketch our graph, and there it is.
Okay, I hope you found this video helpful. Thank you for watching!