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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.)
Hey, everyone. Welcome back. You know, it's funny how some of the most common shapes we see every day, like the arc of a basketball or the curve of a bridge, are actually based on some pretty cool math.
Speaker 2:Absolutely. And that's what we're diving into today. Those graceful curves, those parabolas, and the math behind them, quadratic functions.
Speaker 1:Quadratic functions. It kinda makes you think back to high school algebra. Maybe not always the most exciting memories, But stick with us because today we're gonna uncover the magic behind these equations and how they connect to the real world in some pretty amazing ways.
Speaker 2:It's easy to get lost in the formulas, but what we really wanna do is bring those equations to life, to see them as a way of describing the relationships we see all around us.
Speaker 1:Okay. So let's say you're about to teach quadratic functions and you really want your students to get it, to see how it all connects. Where would you even begin?
Speaker 2:You know, a lot of times, the best place to start is with something familiar, linear equations. Think back to that classic y m x plus b.
Speaker 1:Right. Like, y eight two x. It's a blast from the past. I remember that. It's a line on a graph.
Speaker 1:Right?
Speaker 2:Exactly. And each part of that equation tells us something specific about that line. That 8 feet at the end, that's our y intercept. That's where the line crosses the I axis.
Speaker 1:So, like, it's intercepting the y axis at 8. Got it.
Speaker 2:And then that next d two attached to the x, that's our slope. It tells us how steep the line is. Basically, for every step we take to the right on the x x axis, how far up or down we're going on the y axis.
Speaker 1:So you've got this equation that's basically like a set of instructions for drawing a line, and those instructions are linked to a visual, the graph.
Speaker 2:Precisely. And with quadratic functions, it's the same idea, but with a curve thrown in. We're still describing a relationship visually, but now we're talking about parabolas, not straight lines.
Speaker 1:Okay. Things are getting a little more interesting. Yeah. So how do we go from our trusty y opals mx+b to these curvy quadratic equations.
Speaker 2:Instead of just x, now we'll start seeing x squared, and that's what gives us that characteristic parabolic shape. Let's take a look at an example. Hhosangus60d plus 1 plus 78t plus 10.
Speaker 1:Okay. Walk me through this one. What are we looking at here?
Speaker 2:This equation actually models something you see every day, whether you realize it or not projectile motion. Imagine throwing a ball upwards. This equation describes the height of that ball over time.
Speaker 1:So if we plugged in different times, like one second, 2 seconds, we can figure out exactly where the ball is in the air at that moment.
Speaker 2:Exactly. Now this particular equation is in what we call standard form. But just like with linear equations, we can write it in different ways to emphasize different aspects. Let's look at his factored form. H, 2t5, 8t+.
Speaker 1:Woah. That looks way more complicated.
Speaker 2:It might seem that way at first, but trust me, it's just giving us a different set of clues about this ball's journey. And the cool thing is that both forms, the standard and the factored, they both represent the same function. It's like looking at the same object from different angles, you get a different perspective, but it's still the same object.
Speaker 1:Okay. So we've got these 2 different forms, standard and factor, both describing the same parabolic curve. How does that actually play out on a graph? What are we looking for?
Speaker 2:It's like with the linear equation. We were looking for those visual landmarks, the y intercept and the slope. Well, with quadratics, we have similar landmarks, just in a slightly different way.
Speaker 1:Alright. Let's go back to that factored form, h minus 2 to dash 5, 8 t plus 1. What clues can we get from this form about what our parabola would look like?
Speaker 2:The factored form is great for figuring out those x intercepts or in this case, t intercepts since our variable is t.
Speaker 1:Right. Because it's time.
Speaker 2:Exactly. It's about time, not x. And remember, the intercepts are where the graph crosses the axis. So we're trying to figure out where does our parabola cross that horizontal time axis.
Speaker 1:Which is basically that you're saying when that ball hits the ground.
Speaker 2:Right? When the height is 0. So if we set the entire equation equal to 0, we see that happens at 2 points in time, when t yields 5 and when t negates 18. Those are our t intercepts, the two places where the parabola would intersect that time axis on the graph.
Speaker 1:Okay. That makes sense. But what about that negative 18? How can time be negative? We're talking about a ball being thrown, not time travel.
Speaker 2:You've got a point there. We wanna think about the context. So mathematically, yeah, the parabola extends forever in both directions.
Speaker 1:Infinite parabola.
Speaker 2:Right. But in this scenario, we only really care about the part of the graph where time is positive from the moment that ball is thrown upwards. That negative t intercept, it's like a phantom intercept. It's there mathematically, but it doesn't really have a meaning in our ball throwing scenario.
Speaker 1:So the parabola wants to keep going, but we have to kinda put up some boundaries based on the real world.
Speaker 2:Exactly. Now what about that e intersect? Where does that fit in with our parabola?
Speaker 1:Well, with our linear equation, we just looked at the number by itself, the b. Is it the same deal here?
Speaker 2:It's a similar idea, but in the standard form, not the factored form.
Speaker 1:Okay.
Speaker 2:Yeah. So if we go back to that standard form equation, h equal 1 to get a 16 t twats plus 7 d t plus 10, that high plus 10 at the end, that's our y intercept. It tells us that the ball, it starts its journey 10 feet above the ground.
Speaker 1:So each form, it gives us different pieces of the puzzle. The factored form helps us find those t intercepts where the parabola crosses the horizontal axis, and the standard form gives us that starting height, the y intercept. But why does this even matter? Like, how does understanding this help us in the real world beyond just passing an algebra test?
Speaker 2:Well, imagine you're a company selling video games.
Speaker 1:Right.
Speaker 2:And you've got this revenue equation, r magus 10p dash of 5.
Speaker 1:Okay. So we're switching gears from throwing balls to selling video games, but it's still a parabola. Right?
Speaker 2:Exactly. It's still a parabola. We've just changed our variables.
Speaker 1:Okay. So what are those intercepts telling us in this case? It's not about time anymore. Right?
Speaker 2:Right. Now it's about price. Looking at that factored form, we can see the revenue hits 0 when p equals 25 and when p equals 9.
Speaker 1:So that means if they price the game at 25 or $9, they won't make any money. But why wouldn't they make any money at $9? That seems like a great deal.
Speaker 2:That's where it gets interesting. Remember, revenue isn't just about the price per item. It's also about how many people are actually dying at that price.
Speaker 1:Mhmm.
Speaker 2:At $9, they might sell a ton of games because, hey, it's a steal. But the price is so low, they might not actually be making a profit.
Speaker 1:Oh, right. They might not be able to cover the cost of actually making the game if they sell it that low.
Speaker 2:Exactly. And then on the other hand, at $25, the price might be so high that very few people are willing to buy it, again, leading to low revenue. So the key here is to find that sweet spot, that perfect price point that's gonna bring in the most revenue. And guess what? That's where the vertex of our parabola comes in.
Speaker 1:Okay. I'm intrigued. Tell me more. How do we find that magic price on the graph?
Speaker 2:So the vertex, it's like the peak of the parabola, the highest point on the curve. And because parabolas are symmetrical, that vertex is always right in the middle of those 2 x intercepts.
Speaker 1:So in this case, the vertex would be at a price that's exactly between $9.25.
Speaker 2:You got it. In this case, that'd be a price of $17. That's the price point this model suggests would lead to the highest revenue.
Speaker 1:Wow. Parabolas and profits. Who knew? But you mentioned that this is a model. Right?
Speaker 1:So what are some things that a video game company would need to consider in the real world that might not be captured in this, you know, this elegant symmetrical equation.
Speaker 2:Oh, absolutely. Real world scenarios, they always have those extra layers of complexity. You know, this model we've been looking at, it's assuming a pretty simple relationship between price and revenue. But there's a lot more that can influence those things in real life. Right?
Speaker 2:Things like competition, how how much they're spending on marketing, even just the general hype around a game.
Speaker 1:Right. Like, a a game could be priced perfectly according to the equation, but if nobody's excited to play it, it's probably not gonna sell.
Speaker 2:Exactly. But that's what makes this stuff so fascinating. It gives us this framework, this starting point for understanding these relationships, for making smarter decisions, but it's not the whole picture.
Speaker 1:So it's like the math gives us a really good starting point. It's not gonna tell us everything.
Speaker 2:Right. Exactly. It's a guide, not a guarantee.
Speaker 1:Okay. So we've covered a lot of ground here. We've gone from throwing basketballs to pricing video games, all thanks to the power of quadratic functions.
Speaker 2:And don't forget those bridges, those graceful arcs. That's parabolas at work too.
Speaker 1:That's right. Quadratic functions are everywhere. But before we wrap up, are there any, like, common misconceptions or stumbling blocks that you find students often run into when they're first learning about this stuff?
Speaker 2:Definitely. I mean, one of the big ones is just confusing the roles of standard form and factored form. Like, they're so busy trying to remember how to convert from one to the other that they lose sight of what each form actually tells us about the parabola. You know?
Speaker 1:Yeah. It's like getting so caught up in the steps that you kinda forget what you're actually trying to bake.
Speaker 2:Exactly. And another common mistake is with those signs when you're trying to find the x intercepts from the factored form. It's easy to just glance at an equation like, let's say, y by 3x+5 and think, okay, the intercepts are at positive 3 and positive 5.
Speaker 1:Right. Because you see those numbers in there.
Speaker 2:Yeah. But you gotta remember, it's about what makes each of those factors equal to 0.
Speaker 1:So, it's actually at x equals 3 and x equals negative 5 in that example. It's easy to make that mistake.
Speaker 2:Super easy. And then, another tricky one is the y intercept. Sometimes, students see those numbers in the factored form and they think, oh, I can just multiply those together to get the y intercept, but that's not always how it works.
Speaker 1:Right. Right. It's that constant term all by itself in the standard form that tells us the y intercept.
Speaker 2:Exactly. It all comes back to remembering those key differences between the forms and how they link up to what we see on the graph.
Speaker 1:This has been a fantastic deep dive. I feel like I finally understand what all those x squareds are really up to.
Speaker 2:I'm glad to hear it. The biggest takeaway is just to remember, quadratic functions, they're more than just these abstract equations on a page. Right? They're a way of making sense of the world around us.
Speaker 1:Absolutely. A huge thank you to the authors of illustrative math for providing such rich and engaging material for us to explore. And to you, our amazing listeners, thanks for joining us on this deep dive. Until next time, stay curious.