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A short class with Sarah and her students. The other ways are under More.
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Today, we'll investigate a scenario where a ball is thrown upward. The height of the ball can be modeled by a quadratic equation. Can anyone tell me what a quadratic equation looks like?
Is it something like ax^2 + bx + c = 0?
Exactly! And what's special about the coefficients a, b, and c?
I remember that a should not be zero!
Correct! Now, let's break down our specific equation: h(t) = -5t^2 + 20t + 2. Can anyone identify the values of a, b, and c here?
a is -5, b is 20, and c is 2!
Great job! These values will help us solve for important information about the ball's trajectory.
To find out when the ball hits the ground, we need to set h(t) to zero and solve for t. Can anyone show us how to do that?
We can set -5t^2 + 20t + 2 = 0!
Exactly! Now, let's rearrange it to a standard form, which gives us 5t^2 - 20t - 2 = 0. What’s next?
We can apply the quadratic formula or factor it!
That's right! And remember how we use the quadratic formula: t = (-b ± √(b² - 4ac)) / 2a. Let's plug in our values for a, b, and c.
It gives the time when the ball hits the ground!
Correct! This is a vital application of mathematics in real life.
Now let’s find the maximum height the ball reaches. Does anyone remember how to use the vertex formula?
Isn't it -b/(2a) to find the t-value?
Exactly! Let’s calculate that using our values. What is -b/(2a) in this case?
It's -20/(2*-5) which equals 2 seconds!
Great work! So now that we know the time at which the maximum height is achieved, how do we find that height?
We plug the time back into the height formula, right?
Yes! Let’s calculate h(2) to find the maximum height.
Looking back, what have we learned about using quadratic equations in real-life scenarios such as a ball thrown into the air?
We can find out when it hits the ground and the maximum height!
It’s helpful for understanding motions in physics too!
Absolutely! Quadratic equations are essential for modeling various physical scenarios.
It really connects algebra with real-world applications!
Well said! Let’s keep practicing to reinforce these concepts further.
Engaging students in real-world applications of math for deeper understanding.
Using quadratic equations to model physical phenomena, such as projectile motion.
Determining when the height of a projectile equals zero to find when it touches the ground.
When the ball is thrown so high, remember to calculate, don't be shy! Set h(t) to zero, that’s your cue, to find the time when it hits - it’s true!
Once there was a curious ball named Bouncy who loved to fly. When thrown high, it reached for the sky, but to know when it would fall, it had to answer the quadratic call!
To remember the steps of solving: Set height to zero (hit the ground), use the vertex for max height found!
HIT - Height, Initial velocity, Time; remember this when thinking about projectile motion.
Term
Discriminant
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Term
Vertex
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Term
Maximum Height
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An equation of the form ax^2 + bx + c = 0, where a ≠ 0.
The value D = b^2 - 4ac, which determines the nature of the roots of a quadratic equation.
The highest or lowest point on the graph of a quadratic function; found using the formula t = -b/(2a).
The peak height reached by an object in projectile motion, found using the vertex of the quadratic equation.
The motion of an object thrown into the air affected by gravity, often modeled by a quadratic function.
This section focuses on applying quadratic equations to real-world scenarios, including determining when an object will hit the ground and finding maximum height.
In this section, the importance of understanding quadratic equations in real-world applications is explored. Students are challenged to investigate a scenario involving a ball's height over time using the quadratic equation, prompting a practical understanding of the concepts through hands-on investigation.
In this section, students explore the practical application of quadratic equations through an engaging investigation of a projectile, specifically a ball thrown upward. The height of the ball is described by the quadratic function
h(t) = -5t^2 + 20t + 2,$$ where $h(t)$ represents the height in meters after $t$ seconds. Two primary challenges are presented: determining the time it takes for the ball to hit the ground and calculating the maximum height it achieves during its ascent. By employing the quadratic equation principles, students gain insight into the relationship between algebra and real-world situations, highlighting the significance of quadratic functions in modeling and solving practical problems.