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Interactive Audio Lesson
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Understanding the Quadratic Equation
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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.
Finding When the Ball Hits the Ground
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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.
Calculating Maximum Height
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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.
Connecting the Dots
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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.
Introduction & Overview
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Quick Overview
Standard
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.
Detailed
Inquiry-Based Challenge
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.
Audio Book
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Problem Introduction
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Investigate: A ball is thrown upward with an initial velocity of 20 m/s from a height of 2 m. The height ℎ in meters after 𝑡 seconds is given by:
ℎ(𝑡) = −5𝑡² + 20𝑡 + 2
Detailed Explanation
In this problem, we have a scenario where a ball is thrown into the air with an initial speed of 20 meters per second from a height of 2 meters above the ground. The equation given, ℎ(𝑡) = −5𝑡² + 20𝑡 + 2, is a quadratic equation that helps us calculate the height of the ball (ℎ) at any time (𝑡) seconds after it was thrown. The negative coefficient (-5) indicates that the height of the ball decreases over time due to gravity after peaking.
Examples & Analogies
Imagine throwing a basketball upwards. Initially, it rises quickly but eventually slows down and comes back down, just like the ball in this problem. The quadratic formula predicts the ball's height at any second after it is thrown.
Finding When the Ball Hits the Ground
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- After how many seconds will the ball hit the ground?
(Hint: Set ℎ(𝑡) = 0 to find when it hits the ground)
Detailed Explanation
To determine when the ball hits the ground, we need to find the time 𝑡 when the height ℎ is equal to 0. This involves solving the equation:
−5𝑡² + 20𝑡 + 2 = 0.
We can apply the quadratic formula to find the values of 𝑡 that satisfy this equation. The quadratic formula states that for an equation of the form 𝑎𝑡² + 𝑏𝑡 + 𝑐 = 0, the solutions for 𝑡 can be found using 𝑡 =
(-𝑏 ± √(𝑏² - 4𝑎𝑐)) / (2𝑎). Here, 𝑎 = -5, 𝑏 = 20, and 𝑐 = 2.
Examples & Analogies
Think of tracking a ball thrown upwards. You want to know when it will come back down and touch the ground. Solving this equation is like determining when a runner will cross the finish line based on their speed and starting position.
Finding the Maximum Height
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- What is the maximum height reached?
(Hint: Use vertex formula for max height.)
Detailed Explanation
The maximum height of a projectile like this ball can be found using the vertex formula of a quadratic equation, which states that the vertex (the maximum or minimum point depending on the concavity) occurs at 𝑡 = -𝑏/(2𝑎). In this equation, once again, we have 𝑎 = -5 and 𝑏 = 20. Plugging these values into the formula gives us the time at which the maximum height occurs. After finding this time, we substitute it back into the height equation to find the maximum height itself.
Examples & Analogies
Consider the arc of a thrown frisbee. At some point, it reaches its highest point before coming down again. Using the vertex formula helps you find that peak height just as it helps calculate the ball's maximum height in this case.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Inquiry-Based Learning: Engaging students in real-world applications of math for deeper understanding.
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Quadratic Modeling: Using quadratic equations to model physical phenomena, such as projectile motion.
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Finding Roots: Determining when the height of a projectile equals zero to find when it touches the ground.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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A ball thrown from a height of 2 m with an initial velocity of 20 m/s follows the equation h(t) = -5t^2 + 20t + 2.
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To find out when the ball hits the ground, we solve h(t) = 0, leading to solving the quadratic equation -5t^2 + 20t + 2 = 0.
Memory Aids
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🎵 Rhymes Time
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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!
📖 Fascinating Stories
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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!
🧠 Other Memory Gems
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To remember the steps of solving: Set height to zero (hit the ground), use the vertex for max height found!
🎯 Super Acronyms
HIT - Height, Initial velocity, Time; remember this when thinking about projectile motion.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Quadratic Equation
Definition:
An equation of the form ax^2 + bx + c = 0, where a ≠ 0.
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Term: Discriminant
Definition:
The value D = b^2 - 4ac, which determines the nature of the roots of a quadratic equation.
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Term: Vertex
Definition:
The highest or lowest point on the graph of a quadratic function; found using the formula t = -b/(2a).
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Term: Maximum Height
Definition:
The peak height reached by an object in projectile motion, found using the vertex of the quadratic equation.
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Term: Projectile Motion
Definition:
The motion of an object thrown into the air affected by gravity, often modeled by a quadratic function.