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Lesson by lesson podcasts for teachers of Illustrative Mathematics®.
(Based on IM 9-12 Math™ by Illustrative Mathematics®, available at www.illustrativemathematics.org.)
Ever look at a quadratic equation and wish its graph, you know, would just magically appear, like, with all the important parts labeled.
Speaker 2:Right.
Speaker 1:Well, today, we're diving into the world of quadratic functions, those specifically in factored form
Speaker 2:Yeah.
Speaker 1:To uncover the secrets, kind of, like, hidden within those equations.
Speaker 2:Yeah.
Speaker 1:So get ready to unlock some serious graphing superpowers.
Speaker 2:What's really cool about factored form is is it's almost like a map, you know
Speaker 1:Okay.
Speaker 2:Leading us straight to this crucial information about the graph.
Speaker 1:Okay. I love a good map, Annalie. But, instead of, like, searching for buried treasure, what kind of landmarks are we looking for with quadratics?
Speaker 2:We are looking for those x intercepts Right. In the vertex.
Speaker 1:So think
Speaker 2:of the x intercepts, like, as those points, where the graph crosses the x axis.
Speaker 1:Right. So that would be, like, where the parabola intersects the ground Yeah.
Speaker 2:You know,
Speaker 1:if we were picturing it, like, in the real world.
Speaker 2:Exactly. Exactly. And what's cool is that the factored form kinda gives us a direct route to finding those.
Speaker 1:Oh.
Speaker 2:So, take the equation, like, 2x +8, for example. Each one of those factors there holds a clue.
Speaker 1:Cool. So, okay, so if our factors are by 2 and x plus 8
Speaker 2:Yeah.
Speaker 1:How do we extract those x intercept clues?
Speaker 2:So we have to remember that any point that's on the x axis, right, always has a y coordinate of 0. Right. So, to find the x intercepts, we basically set each of those factors equal to 0 and then solve for x.
Speaker 1:Okay.
Speaker 2:Does that make sense?
Speaker 1:Okay. Yeah. Yeah. So for by 2, we'd get x 2, and then for x +8, we get x +8.
Speaker 2:Right.
Speaker 1:It's like the signs flipped.
Speaker 2:They do. They flip. And that's, you know, that's an important thing for students to recognize. Right. Because we're essentially you know, it's what we're doing is we're finding the zeros of the function.
Speaker 1:Okay.
Speaker 2:So it's those x values that make that entire equation equal to 0.
Speaker 1:Okay.
Speaker 2:And so in other words, you know, we're we're finding the points where the graph intersects that x axis, meaning that y value 0.
Speaker 1:Okay. That makes perfect sense. Yeah. So we found our x intercepts. What about the vertex?
Speaker 2:Yeah.
Speaker 1:Now if we're thinking about it in, like, real world applications, that would be like finding the highest or lowest point of, you know, like a bridge maybe
Speaker 2:Right.
Speaker 1:Or the peak of, like, a projectile's trajectory.
Speaker 2:Yeah. Absolutely. The vertex is it's critical for, you know, really understanding the behavior of a quadratic function. Right. And and you're right.
Speaker 2:It it does represent that, you know, maximum or minimum value, which is, like you said, essential information in a lot of different applications.
Speaker 1:So how do we, how do we locate this all important vertex on our, like, factored form map?
Speaker 2:Well, one of the really cool things about quadratics is that they're symmetrical.
Speaker 1:Okay.
Speaker 2:Which means that there's kind of, like, this invisible line of symmetry right down the middle of them.
Speaker 1:Okay.
Speaker 2:And the vertex sits perfectly on that line
Speaker 1:Right.
Speaker 2:Which means it's also the midpoint between our two x intercepts.
Speaker 1:Okay. That makes sense.
Speaker 2:Right.
Speaker 1:So to pinpoint that vertex's x coordinate
Speaker 2:Yeah.
Speaker 1:We would average those x intercepts. So in our example with 2 zero and minus 80, that would be 2 +nex82, which gives us minus 3.
Speaker 2:Exactly. Oh, okay. And then finding that e coordinate
Speaker 1:Right.
Speaker 2:Is just as just as straightforward. All we do is we substitute that x value, in this case, MEMH 3
Speaker 1:Look down.
Speaker 2:Back into our original equation, and that'll give us our u value.
Speaker 1:Okay. That makes sense.
Speaker 2:Let's, let's maybe try another example just to, like, really solidify this.
Speaker 1:Yeah.
Speaker 2:So imagine we have the equation by 1 by 3.
Speaker 1:Okay. So using what we just learned, I can see that our x intercepts would be 1,013 Right. Because if we set those equal to 0
Speaker 2:Exactly.
Speaker 1:And to locate that vertex, we would find the midpoint of 1 and 3, which is 2. Right. So the x coordinate of our vertex is 2.
Speaker 2:You got it.
Speaker 1:Okay. Cool.
Speaker 2:And how would you find the e coordinate?
Speaker 1:We would simply plug that x value of 2 back into our original equation.
Speaker 2:Okay.
Speaker 1:So that would be Yeah. 212 to the 3, which would give us negative 1. So the vertex for this parabola would be 2 minus 1.
Speaker 2:You got it.
Speaker 1:This is really starting to click.
Speaker 2:Good. I like it when it clicks.
Speaker 1:But, what happens though when we encounter an equation that isn't presented in this nice, neat, factored form?
Speaker 2:That's a great question, and that's something that, you know, we encounter a lot. Right?
Speaker 1:Yeah. I'm sure.
Speaker 2:And so, you know, in those cases, don't worry. There are there are other there are other tools, you know
Speaker 1:Okay.
Speaker 2:That we have at our disposal. So, you know, like the quadratic formula, for instance, or the method of completing the square
Speaker 1:Right.
Speaker 2:You know, especially when we're dealing with, you know, standard form equations.
Speaker 1:So it's like having different maps for different terrains.
Speaker 2:Precisely. And and understanding these core concepts that we're talking about, you know, how that factored form connects to the x intercepts and the vertex.
Speaker 1:Right.
Speaker 2:That builds such a strong foundation for when we do encounter those other scenarios.
Speaker 1:This is incredibly helpful. We're we're starting to see how understanding these connections you know, it's like having a superpower.
Speaker 2:Mhmm.
Speaker 1:Just by glancing at that quadratic equation, we can predict its graph. Yeah. And we can gain those valuable insights into, you know, its behavior. It's like we're unlocking a new level of mathematical fluency.
Speaker 2:That's exactly what we're doing. And it you know, this understanding really it opens up a world of possibilities when it comes to solving real world problems. Right? So, I mean, imagine using these concepts to, you know, model projectile motion
Speaker 1:Right.
Speaker 2:Design efficient satellite dishes, or even, you know, like, optimize the architecture of buildings. I mean, the applications are really endless.
Speaker 1:That's so cool. Yeah. But I have a feeling that even with the best map and compass, we might still encounter a few wrong turns along the way.
Speaker 2:You're absolutely right. You know, even the most, you know, even the most experienced mathematicians. You know? Yeah. We can we can stumble upon these, these, like, common misconceptions.
Speaker 1:Right.
Speaker 2:And it's it's valuable to kind of address them head on. You know, one really common one is, like, assuming that those numbers we see directly in that factored form are those x intercepts
Speaker 1:Oh.
Speaker 2:Without really thinking about, like, the operations involved. You know? Right. So it's it's easy to kind of, you know, get caught up in the speed of things, but that's where those kind of misunderstandings can creep in.
Speaker 1:Yeah. So instead of seeing, like, right by 5 and thinking, boom, x intercept at 5, we need to slow down a little bit and remember to set that factor to 0 and then solve.
Speaker 2:Exactly.
Speaker 1:Okay. Well
Speaker 2:Exactly. And I think, you know, really emphasizing, like, the why behind those flipped signs can make a world of difference for students.
Speaker 1:Right. Like, making it stick instead of just memorizing a rule.
Speaker 2:Totally.
Speaker 1:So what other, what other traps have you noticed students kinda falling into?
Speaker 2:Another big one is, like, relying solely on a table of values
Speaker 1:Okay.
Speaker 2:To reveal everything about the graph, you know, including that vertex or even those x intercepts.
Speaker 1:Right. Yeah. Tables could be deceiving if we're not careful.
Speaker 2:They totally can.
Speaker 1:It's like trying to, like, understand the whole shape of a mountain range by just looking at, like, a a few scattered elevation markers. You know?
Speaker 2:That's a great analogy. Yeah. Because they're you know, tables give us these snapshots, which are helpful.
Speaker 1:Right.
Speaker 2:But they may not always show us that full picture. Right? Especially when we're dealing with, you know, those in between values, those those non integers.
Speaker 1:So a student might see a few points from a table and think that the vertex is at, you know, a specific integer value, when in reality, it's, like, nestled in between those points.
Speaker 2:Precisely. And so we need to kind of encourage students to zoom out a little bit
Speaker 1:Right.
Speaker 2:And look at that bigger picture. And this is where having those, you know, those multiple representations
Speaker 1:Mhmm.
Speaker 2:Becomes really powerful. Powerful.
Speaker 1:So we've got our equation itself. We've got the table of values, and then we've got you know, we can visualize it with an actual graph. Exactly. And each one of those deepens our understanding. Exactly.
Speaker 1:It's like
Speaker 2:each one
Speaker 1:gives us a different lens through which to view and understand the
Speaker 2:function.
Speaker 1:Okay. So let's let's brainstorm some strategies. If our goal is to kind of guide students away from these pitfalls and toward that deeper understanding, what are some practical things that teachers can, you know?
Speaker 2:Totally. Yeah. So for for tackling that first one
Speaker 1:Okay.
Speaker 2:You know, kinda mixing up the numbers in the factor form with those actual x intercepts
Speaker 1:Right.
Speaker 2:I love to kind of encourage a bit of, like, playful testing.
Speaker 1:K.
Speaker 2:Like, you know, have students plug in the values that they think are those x intercepts and see what happens.
Speaker 1:Oh, I love that.
Speaker 2:Right.
Speaker 1:So if they see, you know, like, my 5 and they assume that it's at x for 5, encourage them to plug 5 back into that original equation.
Speaker 2:Exactly. It's like saying, okay. Let's let's see if this actually works.
Speaker 1:Right. Right.
Speaker 2:And and they'll very quickly see that, like, oh, the output isn't 0.
Speaker 1:Okay.
Speaker 2:And that can spark some really cool
Speaker 1:Yeah.
Speaker 2:Moments as they realize, oh, wait. I need to actually solve for x, like, setting each of those factors equal to 0.
Speaker 1:Turn those mistakes into learning opportunities.
Speaker 2:Yeah. Exactly. Yeah. Exactly.
Speaker 1:Okay. What about that second misconception where tables might not be, you know, revealing the complete story? How can we how can we help students kind of see beyond the dots, so to speak?
Speaker 2:Yeah. And this is where, you know, I think graphing technology can be a real game changer.
Speaker 1:Okay.
Speaker 2:Right? So, you know, using tools like Desmos, for example, allows students to, like, really visualize that entire function, you know, moving beyond just those, like, limited integer values in a table.
Speaker 1:Right. So instead of just plotting, like, you know, 5 or 6 points, they can see that elegant curve of the parabola
Speaker 2:Exactly.
Speaker 1:And get a much better sense of where that vertex actually lies
Speaker 2:Yeah.
Speaker 1:Even if it's, you know, not at 2 it's at 2.25 or something like that.
Speaker 2:Exactly. And I think, you know, being able to actually visualize that can you know? Yeah. It just leads to a much, like, deeper and intuitive understanding. But Good.
Speaker 2:You know, and this is key, I think, is that we have to, you know, we have to emphasize that, like, the technology should be a tool for exploration and confirmation, not necessarily a replacement for, like, really grasping those fundamental concepts. Right. It's
Speaker 1:like it's like having, you know, a really fancy GPS system.
Speaker 2:Right.
Speaker 1:Like, incredibly useful. Sure. But we still need to know how to read a map Right. And understand those kind of directional cues Yeah. To really navigate.
Speaker 2:I like that. I like that a lot. And and it's the same with, you know, with quadratic functions as well. And, you know, I think we've we've made some good progress today uncovering, you know, those connections between that factored form, the graph Right. And and those those common misconceptions.
Speaker 2:And Sounds
Speaker 1:like we're not just, you know, learning the moves, but, like, understanding the strategy behind the game of algebra.
Speaker 2:Exactly. And speaking of strategy, what if we what if we took all this knowledge Okay. And and flip the script a little bit?
Speaker 1:Oh, I like where you're going with this.
Speaker 2:Yeah. So
Speaker 1:we've been, like, really digging deep into how to, you know, glean these insights from the factored form.
Speaker 2:Right.
Speaker 1:But could we could we work backward? Yeah. Could we start with a graph? Yeah. Pinpoint those key landmarks Yeah.
Speaker 1:Like the x intercepts and vertex
Speaker 2:Yep.
Speaker 1:And then use that to, like, reverse engineer the equation in factored form.
Speaker 2:That's a fantastic question. And, you know, it it gets at the heart of, like, true mathematical thinking there.
Speaker 1:Right.
Speaker 2:And and the answer is a resounding yes.
Speaker 1:Yes.
Speaker 2:Once we understand, you know, the the relationships between that equation and the graph Right. We can use those patterns, you know, to work in both directions.
Speaker 1:That's incredible. It's like we're not just, like, map readers now. We're Yeah. We're mapmakers.
Speaker 2:Exactly. We're we're developing this, like, fluency that allows us to not just interpret these mathematical representations Right. But to create our own.
Speaker 1:This has been an amazing journey. We've gone from, you know, maybe feeling a little bit lost in the weeds of quadratic equations
Speaker 2:Right.
Speaker 1:To, like, confidently navigating their graphs and even, you know, reverse engineering equations from those visual representations.
Speaker 2:It's all about connecting those dots. You know? Seeing those patterns and really understanding, you know
Speaker 1:Right.
Speaker 2:The why behind the how.
Speaker 1:Absolutely. A huge thank you to the authors of Illustrative Math for providing such insightful resources.
Speaker 2:Yes. Thank you.
Speaker 1:And to our listeners, you know, keep exploring, keep questioning, and, you know, most importantly, keep those mathematical sparks flying.