SAT Prep: Test 5 Section 9 Part 4

We're on problem 14.

If the function f as defined
by f of x is equal to x

squared plus bx plus c, where
b and c are positive

constants, which of the
following could be

the graph of f?

So b and c are positive.

So what does that tell us?

So a couple of things you
immediately know.

The x squared term-- whatever's
the coefficient of

x squared, that tells us whether
the parabola opens up

or opens down.

It opens up if the coefficient
on the x term is positive,

which it is.

It's positive 1, right?

So we know the graph is
going to open up.

It's not going to be a

downward-opening graph like that.

So we know the graph is going
to look like a U, as opposed

to-- I don't know what
this is like.

An A without the line?

Whatever.

So we know it's going
to be a U.

And what else do we
know about it?

What do we know about
its y-intercept?

Well, this is the y-intercept.

When x is equal to 0, f
of 0 is equal to what?

These two terms are going to be
0, so f of 0 is equal to c.

And so we know that the
y-intercept is positive.

So let's see if that
alone allows us

to solve the problem.

So we know that the y-intercept
is positive, so

when f of 0 is going to
intersect someplace on the

positive y-axis.

And we know that it's an
upward-opening graph.

So the graph could
look like this.

It could also look like-- well,
this b term-- I won't

get too much into the b term.

So we actually know that it
is shifted to the left.

But based on just what I said,
that it's opening upwards and

that it has a positive
y-intercept-- this would've

also been a legitimate graph.

Anything that is opening upwards
like a U-- because

this is a positive x squared
here-- and intersects the

y-axis in the positive area
would be a correct answer.

And if you look at the choices,
A opens down.

That's not the right answer.

B opens down.

Not the right answer.

C intercepts the y-axis
at 0, so that's

not the right answer.

D opens up, but it intersects
the y-axis in the negative y.

And then finally E is
very similar to

what I drew in yellow.

So that is our answer.

E.

Next problem.

Almost finished with
this section.

OK, they drew us a cube.

Let me draw it big, because
this looks like it might

involve some fancy-- let
me draw the cube.

So this is the front face.

Whoops.

Undo.

All my time is spent drawing
diagrams. Cube.

Drawing a diagram.

And then it comes
back like that.

Goes back like that.

I should always draw the back
first. Good enough.

I think you get the point.

And it draws the dotted line.

Maybe I'll draw that later,
if I have to.

So they tell us this is point
A, and that this right

here is point B.

They say the cube shown above
has edges of length 2.

So each of these sides are 2.

2, 2, 2, 2, and so on.

And A and B are midpoints
of the two edges.

So A and B are midpoints.

So this side is equal
to this side.

So if the whole side is equal
to 2, and A is a midpoint,

then we know that this
is 1, this is 1, this

is 1, this is 1.

What is the length of AB?

So this is what they want
us to figure out.

They want me to draw it in
kind of a darker color.

No, that's not a darker--
that's that line.

This is a pure visualization
problem.

And at first, you're like, boy,
that's three dimensions.

It's crazy.

What am I going to do?

What you're going to have to
do is do the Pythagorean

Theorem twice.

So what do we know?

Well, we know everything we need
to solve this problem.

Let's see if we can do it.

So let me draw that bottom
surface of the cube.

So I'll ask you a question.

Can we figure out-- so
let me draw a line

along the bottom surface.

Let's say that this line I'm
drawing is along the bottom

surface of the cube.

So I'm going to draw
this line.

That's along the bottom
surface of the cube.

And let me label this
side like this.

This is actually a really
fun problem.

I'll do this in yellow.

So can we figure out what
that magenta dotted

line's length is?

Well, sure.

We know, first of all, that
is a right triangle,

the way I drew it.

I can redraw it like this, where
this is yellow, brown,

and then dotted line magenta.

All I did is I kind of
flipped it up so that

you could see it.

So what is that?

We know the brown length.

We know that's 1.

We know the yellow
length is what?

It's 2.

Because all the sides
of the cube are 2.

So we can use the Pythagorean
Theorem to figure out the

magenta line.

So it's 1 squared plus
2 squared is equal

to this side squared.

So we can just take the
square root of it.

So that equals the square
root of 5, right?

1 plus 4.

Square root of 5 is
this magenta line.

So now can we figure
out that gray line

that I drew at first?

Well, sure.

Because now we have another
right triangle.

That gray line is just the
hypotenuse of this right

triangle, that's kind
of at an angle.

So let me make sure you
understand what I'm saying.

If I were to flatten
it out, I have this

green side right here.

I have this dotted line base,
and then I have the gray line,

which is the hypotenuse.

And just so you know, this point
right here would be A,

and then this point right
here would be B.

And do we know the sides?

Yes, we do.

We know this green line here is
1, so this side here is 1.

I'm just redrawing it here.

We know the magenta dotted line
at the bottom, which I

kind of switched colors a bit.

We just figured out that that
magenta bottom line is

square root of 5.

So now we can just use the
Pythagorean Theorem to figure

out the length of this line.

The length of this line squared
is going to be equal

to the square root of 5,
squared, plus 1 squared.

So the hypotenuse squared is
equal to this, so we could

take the square root of that.

So that equals the square
root of-- what's the

square root of 5 squared?

Well, it's just 5, right?

Plus 1.

So it's just equal to the
square root of 6.

So this line is equal to
the square root of 6.

And that is choice D.

We luckily have not
made a mistake.

Last problem.

Problem 16.

Let x-- with this kind of
oval-looking thing around it--

be defined as x squared minus
x, for all values of x.

If a-- so they're telling us
that a is equal to a minus 2,

what is the value of a?

So what does this mean? a
kind of with this oval?

Well, that just means that
a squared minus a, right?

That maps to that,
just like that.

And what does a minus
2 map to?

And I'm just bringing
the equal sign down.

a minus 2 maps to-- every place
where I see an x, I put

an a minus 2 in there.

So that equals a minus 2
squared, minus a minus 2.

And now we just keep solving.

And most of the solving is
on the right-hand side.

So let's see.

That equals a squared minus
4a plus 4 minus a plus 2.

So we get a squared minus
a is equal to a

squared minus 5a, right?

Minus 4a minus a plus 6.

We can subtract a squared
from both sides.

Let me add 5a to both sides.

So 5a, so you get 4a
is equal to 6.

Divide both sides by 4.

You get a is equal to 6/4, which
equals 1 and 1/2 or 3/2.

And that is choice C.

And we are done this section.

I'll see you in the next
practice test.