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Solving Quadratic Equations by Graphing
相關課程
- What I want to do in this video is to actually use a
- graphing calculator, get some experience on the calculator
- itself, for graphing some of these quadratics and actually
- figuring out the roots visually by
- looking at those graphs.
- So the first equation I want to find the x-intercept for is
- y is equal to negative 3x squared plus 4x minus 1.
- So we're just going to go straight to the calculator.
- We'll get our TI-85 out here.
- And there are obviously better things you can use, especially
- on your computer, that are going to be much more visually
- interesting and in color and even faster, but many of us
- have a TI-85 at our disposal.
- The TI-85 hasn't changed much actually since I was in high
- school, or the TI-83 or whatever else there is out
- there, TI-92.
- And this at least will give you a sense of how to use it,
- because this is what you have, and how to analyze the graph
- once you are able to graph it.
- So let's-- I already turned it on-- so let's input this graph
- right here.
- So we click on GRAPH and then we'll click on-- this F1 tells
- us to pick this option right here, this y of x, right?
- So they're saying y is a function of x, y is indeed a
- function of x.
- So let's click there, and then that's going to be negative
- 3x-- let's see where is the variable--
- there's the x variable.
- x squared plus 4x, minus 1.
- There is our function.
- Now let's graph it.
- So we want to pick this choice up here.
- So to pick this second choice-- so notice, if you
- want to pick any of these choices on the bottom row, you
- can just punch in these numbers.
- But if you want to pick the choice that's above, you could
- press 2nd, and then press that, and then 2nd tells you
- to pick the thing that's right above that.
- so I'll pick the thing in the yellow, essentially.
- So I'll do 2nd.
- And I'm selecting the GRAPH option.
- When I was in high school, this was all very fancy stuff,
- but now compared to pretty much most cell phones, this is
- pretty archaic.
- But it gets the job done.
- Now, let's zoom in, because I really can't see
- the 0's that well.
- So let me zoom here.
- I like to use the box, we zoom in on a certain area that I
- can define.
- So, I'll select the box, and then I select where I want to
- start the box for using the arrow keys.
- Press ENTER, that tells it to start the box, and then I can
- zoom in using the box.
- I define the box, and when I have a box I
- like, I press ENTER.
- And it'll redefine the range of x and y values that it
- calculates the graphing function.
- Now, I want to figure out what these x-intercepts are.
- So I'm going to use the trace function.
- And the trace function is just going to
- walk along this graph.
- So let me hit 2nd, trace.
- And now I'm walking along this graph as I go to the right.
- It just increments the x-value.
- We're getting closer and closer to y is equal to 0.
- And it looks like it happens right at about 1.
- Right as I cross 1, I'm crossing y is equal to 0.
- y is a very small positive number here, and then it goes
- to becoming a less small, but still a small negative number,
- right as we cross 1.
- I'm going to evaluate it directly at x is equal to 1.
- But, I think that's a pretty good approximation, and we can
- even try it out in our equation.
- Let's try it out.
- If x is equal to 1-- let me write it this way.
- y of 1, if we write y is a function of 1, this is going
- to be equal to negative 3 times 1 squared, plus 4 times
- 1, minus 1.
- So this is equal to negative 3 plus 4, minus 1,
- which is indeed 0.
- So the point x is equal to 1, y is equal to 0 is definitely
- on the graph, and this is one of our x-intercepts.
- Let's see if we can figure out the other one using our
- calculator.
- So let's see, we need to figure out this other point
- right here.
- Let's get the zoom box going.
- Actually, let's try just to trace there.
- Scrolling all the way.
- And it looks like it's around 0.33.
- I suspect it's close to 1/3, but let me zoom in a little
- bit better.
- So let me click GRAPH again, and click on zoom.
- I'm going to use the box again.
- And then let me do the top left corner of the bottom, and
- zoom in really narrowly on that point right there.
- Really, really zoom in.
- So it's around 0.33 that I want to zoom in.
- So that's going to be the beginning of my box, and let
- me go right below the y-axis and then go right above 0.33.
- And so it's really zoomed in on that
- y-intercept right there.
- So now if I trace it, I really should be able to figure out
- what that value is.
- So let me do 2nd, and trace, and keep scrolling.
- So it's a little bit more than 0.33.
- It looks like it's really approaching
- 0.3333333, which is 1/3.
- It looks like it's 0.3 repeating.
- Then we're getting to very, very small y-values.
- So it looks like 1/3, so let's try out 1/3 and see if we got
- the right answer.
- So, if we have y of 1/3 is equal to negative 3 times 1/3
- squared, plus 4, times 1/3, minus 1.
- This is equal to-- negative 3 times 1/9 is negative 1/3.
- This is plus 4/3, and then minus 1 or negative 1,
- depending on how you view it, it's the same thing as
- minus 3 over 3.
- So this is going to be negative 1 plus 4, minus 3,
- all of that over 3, which is indeed 0.
- So we also have the x-intercept.
- x is equal to 1/3, y is equal to 0.
- Let's do another one.
- Let's say we have y is equal to 1/2x squared
- minus 2x, plus 3.
- Let's see if we can get its roots, or the 0's, of this
- equation, where it intersects the x-axis.
- So let's get our calculator back.
- And we're going to have to zoom out of this thing.
- So let me do zoom.
- I think I can just use the previous zoom.
- Well, the previous zoom is still a little bit
- not too zoomed in.
- Let me just do the previous zoom before that.
- And then, I'll do another zoom-- oh, it just keeps
- alternating between the zooms, so maybe I have to clear out.
- I am not an expert at this.
- All right.
- So let me just input my new graph.
- So it's 1/2x squared.
- I could write that as 0.5x squared minus 2x, plus 3.
- And then I want to graph this.
- Oh, so it's on that super-zoomed in version, so
- let me zoom out.
- There you go, zoom standard, that resets it.
- There you go.
- That's all I had to do, the standard zoom.
- So you see.
- Let's see, where does it intersect the x-axis?
- Well, it doesn't intersect the x-axis.
- It just goes down here, then keeps going up.
- So this quadratic function right here will have no
- x-intercepts.
- I mean, you can see it graphically here.
- It does not intersect the x-axis.
- So, if you were trying to solve the equation 1/2x
- squared minus 2x, plus 3 is equal to 0, what you're
- essentially trying to do is say, what x-values give me y
- is equal to 0 here?
- We just graphed it, it never intersected the x-axis.
- It just did something like this.
- It never equaled y is equal to 0.
- So this right here will have no real solutions.
- And I use the word real in the mathematical sense.
- There no real number solutions.
- We'll see, not too far off in the future, that there are
- things called complex numbers, and those can be solutions to
- quadratics that don't intersect the x-axis, at least
- in kind of the real domain.
- Now let's do one more of these.
- Let's say we have x squared plus 6x, plus 9 is equal to 0.
- And we should actually be able to factor this in our head.
- What two numbers add up to 6 and when you multiply
- them you get 9?
- It probably jumps out of your head.
- It's 3 and 3.
- So this is x plus 3, times x plus 3 is equal to 0.
- Or-- and we just have to use one of these-- x plus 3 is
- equal to 0.
- Subtract 3 from both sides. x is equal to negative 3.
- So this has one 0.
- So how do you think this is going to look if it only
- intersects the x-axis at exactly one point?
- Well, the best guess is is that that one point is going
- to be the vertex.
- So let's graph it just to verify for ourselves.
- So let me to clear this one out, and we have x squared
- plus 6x, plus 9.
- And we want to graph it.
- And there you go.
- The vertex is right at x is equal to negative 3, y is
- equal to 0.
- It just hits it right over there.
- So our intuition was correct.
- And just as a bit of review-- and I should have pointed this
- out when I was showing you the other graphs-- notice, this
- has a positive coefficient on x-squared.
- It is upward-opening.
- The second problem we did also had a positive coefficient.
- It didn't intersect the x-axis, but it was also
- upward-opening.
- And if you rewind this video to the very first problem we
- did, you saw it had a negative coefficient, and it also had a
- downward-opening parabola, or downward-opening u-shape.
- Well, anyway, hopefully you found this little practice
- with the graphing calculator helpful.
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