Examples: Graphing and interpreting quadratics 36-38, shifting quadratic graphs and finding x-intercepts (roots)
Examples: Graphing and interpreting quadratics
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- 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.
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