Example: Parabola vertex and axis of symmetry Quadratic Functions 2
Example: Parabola vertex and axis of symmetry
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- We need to find the vertex and the axis of
- symmetry of this graph.
- The whole point of doing this problem is so that you
- understand what the vertex and axis of symmetry is.
- And just as a bit of a refresher, if a parabola looks
- like this, the vertex is the lowest point here, so this
- minimum point here, for an upward opening problem.
- If the parabola opens downward like this, the vertex is the
- topmost point right like that.
- It's the maximum point.
- And the axis of symmetry is the line that you could
- reflect the parabola around, and it's symmetric.
- So that's the axis of symmetry.
- That is a reflection of the left-hand side along that axis
- of symmetry.
- Same thing if it's a downward-opening parabola.
- And the general way of telling the difference between an
- upward-opening and a downward-opening parabola is
- that this will have a positive coefficient on the x squared
- term, and this will have a negative coefficient.
- And we'll see that in a little bit more detail.
- So let's just work on this.
- Now, in order to figure out the vertex, there's a quick
- and dirty formula, but I'm not going to do the formula here
- because the formula really tells you nothing about how
- you got it.
- But I'll show you how to apply the formula at the end of this
- video, if you see this on a math test and just want to do
- it really quickly.
- But we're going to do it the slow, intuitive way first.
- So let's think about how we can find either the maximum or
- the minimum point of this parabola.
- So the best way I can think of doing it is to
- complete the square.
- And it might seem like a very foreign concept right now, but
- let's just do it one step at a time.
- So I can rewrite this as y is equal to-- well, I can factor
- out a negative 2.
- It's equal to negative 2 times x squared minus 4x minus 4.
- And I'm going to put the minus 4 out here.
- And this is where I'm going to complete the square.
- Now, what I want to do is express the stuff in the
- parentheses as a sum of a perfect square and then some
- number over here.
- And I have x squared minus 4x.
- If I wanted this to be a perfect square, it would be a
- perfect square if I had a positive 4 over here.
- If I had a positive 4 over there, then this would be a
- perfect square.
- It would be x minus 2 squared.
- And I got the 4, because I said, well, I want whatever
- half of this number is, so half of negative
- 4 is negative 2.
- Let me square it.
- That'll give me a positive 4 right there.
- But I can't just add a 4 willy-nilly to
- one side of an equation.
- I either have to add it to the other side or I would have to
- then just subtract it.
- So here I haven't changed equation.
- I added 4 and then I subtracted 4.
- I just added zero to this little expression here, so it
- didn't change it.
- But what it does allow me to do is express this part right
- here as a perfect square. x squared minus 4x plus 4 is x
- minus 2 squared.
- It is x minus 2 squared.
- And then you have this negative 2 out front
- multiplying everything, and then you have a negative 4
- minus negative 4, minus 8, just like that.
- So you have y is equal to negative 2 times this entire
- thing, and now we can multiply out the negative 2 again.
- So we can distribute it.
- Y is equal to negative 2 times x minus 2 squared.
- And then negative 2 times negative 8 is plus 16.
- Now, all I did is algebraically
- arrange this equation.
- But what this allows us to do is think about what the
- maximum or minimum point of this equation is.
- So let's just explore this a little bit.
- This quantity right here, x minus 2 squared, if you're
- squaring anything, this is always going to
- be a positive quantity.
- That right there is always positive.
- But it's being multiplied by a negative number.
- So if you look at the larger context, if you look at the
- always positive multiplied by the negative 2, that's going
- to be always negative.
- And the more positive that this number becomes when you
- multiply it by a negative, the more negative this entire
- expression becomes.
- So if you think about it, this is going to be a
- downward-opening parabola.
- We have a negative coefficient out here.
- And the maximum point on this downward-opening parabola is
- when this expression right here is as small as possible.
- If this gets any larger, it's just multiplied by a negative
- number, and then you subtract it from 16.
- So if this expression right here is 0, then we have our
- maximum y value, which is 16.
- So how do we get x is equal to 0 here?
- Well, the way to get x minus 2 equal to 0-- so
- let's just do it.
- x minus 2 is equal to 0, so that happens when
- x is equal to 2.
- So when x is equal to 2, this expression is 0.
- 0 times a negative number, it's all 0, and then y is
- equal to 16.
- This is our vertex, this is our maximum point.
- We just reasoned through it, just looking at the algebra,
- that the highest value this can take on is 16.
- As x moves away from 2 in the positive or negative
- direction, this quantity right here, it might be negative or
- positive, but when you square it, it's going to be positive.
- And when you multiply it by negative 2, it's going to
- become negative and it's going to subtract from 16.
- So our vertex right here is x is equal to 2.
- Actually, let's say each of these units are 2.
- So this is 2, 4, 6, 8, 10, 12, 14, 16.
- So my vertex is here.
- That is the absolute maximum point for this parabola.
- And its axis of symmetry is going to be along the line x
- is equal to 2, along the vertical line x is equal to 2.
- That is going to be its axis of symmetry.
- And now if we're just curious for a couple of other points,
- just because we want to plot this thing, we could say,
- well, what happens when x is equal to 0?
- That's an easy one.
- When x is equal to 0, y is equal to 8.
- So when x is equal to 0, we have 1, 2, 3, 4-- oh, well,
- these are 2.
- 2, 4, 6, 8.
- It's right there.
- This is an axis of symmetry.
- So when x is equal to 3, y is also going to be equal to 8.
- So this parabola is a really steep and narrow one that
- looks something like this, where this right here is the
- maximum point.
- Now I told you this is the slow and intuitive way to do
- the problem.
- If you wanted a quick and dirty way to figure out a
- vertex, there is a formula that you can derive it
- actually, doing this exact same process we just did, but
- the formula for the vertex, or the x-value of the vertex, or
- the axis of symmetry, is x is equal to negative b over 2a.
- So if we just apply this-- but, you know, this is just
- kind of mindless application of a formula.
- I wanted to show you the intuition why this formula
- even exists.
- But if you just mindlessly apply this, you'll get--
- what's b here?
- So x is equal to negative-- b here is 8.
- 8 over 2 times a.
- a right here is a negative 2.
- 2 times negative 2.
- So what is that going to be equal to?
- It is negative 8 over negative 4, which is equal to 2, which
- is the exact same thing we got by reasoning it out.
- And when x is equal to 2, y is equal to 16.
- Same exact result there.
- That's the point 2 comma 16.
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