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Current time:0:00Total duration:11:31

We're on problem 36. It says which of the following
sentences is true about the graphs of y is equal to 3 times
x minus 5 squared plus 1 and y equals 3 times x plus
5 squared plus 1? So let's do something very
similar to what we did in the past. If you think about it,
both of these equations, y is going to be 1 or greater. Let's just analyze this
a little bit. This term right here, since
we're squaring, is always going to be positive. Even if what's inside the
parentheses becomes negative, if we have x is minus 10, inside
the parenthesis becomes negative, but when
you square it, it always becomes positive. You're going to multiply 3 times
a positive number, so you're going to get
a positive number. So the lowest value that
this could be is 0. The lowest value that y could
be is actually 1. The same thing here, this
number can become very negative, but when you
square it, it's going to become positive. So this expression with the
squared here is going to be positive, and you multiply
it by 3, and it's going to be positive. So the lowest value here is
always going to be zero when you include this whole term. So similarly, the lowest
value y can be is 1. I just want to think about it
a little bit just give you a little bit of an intuition. Let's think of this in the
context of what we learned last time with the shifting. So let me draw it in a color
that you can see. So if that is the y-axis, and
I'll just draw mainly the positive area, so if I were to
just draw y is equal to x squared plus 1, it would look
like this where this is 1. That's y is equal to 1. The graph would look something
along this. y is equal to x squared
plus 1. That's a horrible drawing. Normally, I wouldn't redo it,
but that was just atrocious. y is equal to x squared plus 1
looks something like that. It's symmetric. You get the idea. You've seen these parabolas
before. This is y is equal to
x squared plus 1. Now, if we were to do x minus
5 squared plus 1, what happens to it? Well, let me think about it. What is 3x squared plus 1? Well, then it just increases
a little bit faster. So if I were say y equals 3x
squared plus 1, it might look something like this. It'll just increase a little
bit faster, three times as fast actually. So that would be 3x
squared plus 1. The rate of increase in both
directions just goes faster because you have this constant
term 3 out there multiplying the numbers. OK, now what happens
when you shift it? So let's do x minus 5. So where x equals 0 was the
minimum point before, now if we substitute a 5 here, that'll
be our minimum point. Because then that whole
term becomes zero. So this vertex will now be
shifted to the right. Let me do it in another color. So if this is the point 5, now
this would be the graph. If you just took this graph and
you shifted it over to the right by 5-- I won't draw the
whole thing-- that graph right there would be 3 times x
minus 5 squared plus 1. Remember, the y shift
is always intuitive. If you add 1, you're
shifting it up. If you subtract 1, you're
shifting it down. The x shift isn't. We subtracted 5, x minus 5. We replaced x with x minus 5,
but we shifted to the right. The intuition is there, because
now plus 5 makes this expression zero. So that's 3x minus 5 squared. In the same logic, 3 times x
plus 5 squared is going to be to here, plus 1. That's going to be shifted
to-- let me pick a good color-- to the left. This is going to look
something like this. It's going to be this blue graph
shifted to the left. This is minus 5. So this is the graph right
here of 3 times x plus 5 squared plus 1 Now, hopefully, you
have an intuition. So let's read their statements
and see which one makes sense. Which of the following
is true? Their vertices are maximums.
No, that's not true of any of these. Because the vertices is that
point right there. It's actually the
minimum point. A maximum point would look
something like that. We know that, because you
just go positive. This term can only
be positive. If this was a negative 3, then
it would flip it over. So it's not choice A. The graphs have the same shape
with different vertices. Yeah, both of these graphs have
the shape of 3x squared, but 1 vertices is 10 to the
left of the other one. So I think B is our choice. Let's read the other ones. The graphs have different
shapes with different vertices. No, they have the same shape. They definitely have
the same shape. They both have this
3x squared shape. One graph has a vertex that is
a maximum, while the other has-- no, that's not right. They both are upward facing,
so they both have minimum points. So it's choice B. Next problem, problem 37. Let me see what it says. What are the x-intercepts? Let me copy and paste that. OK, I'll paste it there. What are the x-intercepts
of the graph of that? Well, the x-intercepts, whatever
this graph looks like, I don't know exactly
what it looks like. This graph is going to look
something like this. I actually have no idea
what it looks like until I solve it. It's going to look something
like this. When they say x-intercepts,
they're like, where does it intersect the x-axis? So that's like there
and there. I don't know if those are the
actual points, right? To do that, we set the function
equal to zero, because this is the point
y is equal to 0. You're essentially saying when
does this function equal zero because that's the x-axis
when y is equal to 0. So you set y is equal to 0, and
you get 0 is equal to 12x squared minus 5x minus 2. Whenever I have a coefficient
larger than 1 in front of the x squared term, I find that very
hard to just eyeball and factor, so I use the
quadratic equation. So negative B, this is the B. B is minus 5. So negative negative
5 is plus 5. Negative B plus or minus the
square root of B squared, negative 5 squared is 25, minus
4 times A, which is 12, times C, which is minus 2. So let's just make that
times plus 2 and put the plus out there. A minus times a minus
is a plus. All of that over 2A, all of
that over 24, 2 times A. So that is equal to 5 plus or
minus the square root-- let's see, it was 25 plus 4
times 12 times 2. Because that was a minus 2,
but we had a minus there before, so 8 times 12, so
96, all of that over 24. What's 25 plus 96? It's 121, which is 11 squared. So this becomes 5 plus
or minus 11 over 24. Remember, these are the x-values
where that original function will equal zero. It's always important
to remember what we're even doing. So let's see, if x is equal to
5 plus 11 over 24, that is equal to 16/24, which
is equal to 2/3. That's one potential
intercept. So maybe that's right here. That's x is equal to 2/3
and y is equal to 0. The other value is x is equal
to 5 minus 11 over 24. That's minus 6/24, which is
equal to minus 1/4, which could be this point. I actually drew the graph
not that far off of what it could be. So this would be x is
equal to minus 1/4. Those are the x-intercepts
of that graph. So 2/3 and minus 1/4 is
choice C on the test. We have time for at
least one more. Oh boy, they drew us all
these this graphs. Let me shrink it. I want to be able to
fit all the graphs. So let me copy and paste
their graphs. So this is one where the
clipboard is definitely going to come in useful. OK that's good enough. I've never done something
this graphical. So the graph they say is y is
equal to minus 2 times x minus 1 squared plus 1. So that's what we have
to find the graph of. So immediately when you look
at it, you say, OK, this is like the same thing as y is
equal to minus 2x squared plus 1, but they shifted the x. They shifted the x to
the right by 1. I know it says a minus 1,
but think about it. When x is equal to positive
1, this is equal to 0. So it's going to be shifted
to the right by 1, plus 1. We know that. We know that it's going to be
shifted up by 1, so up plus 1. Then we have to think
is it going to be opening upwards or downwards? Think of it this way: If this
was y is equal to 2x squared plus 1, then this term would
always be positive. It'll just become more and
more positive as you get further and further away from
zero, so it would open up. But if you put a negative number
there, if you say y is equal to minus 2x squared
plus 1, then you're going to open downward. You're just going to get more
and more negative as you get away from your vertex. So we're shifted to the right
by 1, we're shifted up by 1, and we're going to be
opening downwards. So if we look at our
choices, only these two are opening downwards. Both of them are shifted
up by 1. Their vertex is at
y is equal to 1. But this is shifted 1 to
the right and this is shifted 1 to the left. Remember, we said it was
x minus 1 squared. So the vertex happens
when this whole expression is equal to zero. This whole expression is equal
to zero when x is equal to positive 1. So that's right here. So it's actually choice C. When your shifting graphs, that
can be one of the hardest things to ingrain. But I just really encourage
you to explore graphs, practice with graphs with your
graphing calculator and really try to plot points and try to
get a really good grasp of why when you go from minus 2x
squared plus 1 to minus 2 times x minus 1 squared, why
when you replace an x with an minus 1, why this shifts the
graph to the right by 1. Anyway, I'll see you
in the next video.