Using Equations Of Parabolas Quick CheckJackie, Who Is 5 Feet Tall, Throws A Football Into The Air. The Path Of The Football Can Be Expressed By The Equation Y = − 0.05 X 2 + 0.75 X + 5 Y = -0.05x^2 + 0.75x + 5 Y = − 0.05 X 2 + 0.75 X + 5 . Solve To Determine How Far From Jackie The Ball Landed.A.

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Introduction

In mathematics, parabolas are a fundamental concept that can be used to model various real-world phenomena, such as the trajectory of a projectile. In this article, we will explore how to use equations of parabolas to solve problems involving the motion of objects. We will use the example of a football thrown by Jackie to demonstrate how to apply these concepts.

The Equation of a Parabola

A parabola is a quadratic function that can be expressed in the form:

y = ax^2 + bx + c

where a, b, and c are constants. The graph of a parabola is a U-shaped curve that opens upwards or downwards, depending on the value of a.

The Path of the Football

In the example given, the path of the football can be expressed by the equation:

y = -0.05x^2 + 0.75x + 5

This equation represents the height of the football above the ground as a function of the horizontal distance from Jackie.

Solving for the Landing Point

To determine how far from Jackie the ball landed, we need to find the value of x when y = 0. This is because the ball will land on the ground when its height above the ground is zero.

We can set up an equation using the given equation:

0 = -0.05x^2 + 0.75x + 5

To solve for x, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = -0.05, b = 0.75, and c = 5. Plugging these values into the formula, we get:

x = (-(0.75) ± √((0.75)^2 - 4(-0.05)(5))) / (2(-0.05))

Simplifying the expression, we get:

x = (-0.75 ± √(0.5625 + 1)) / (-0.1)

x = (-0.75 ± √1.5625) / (-0.1)

x = (-0.75 ± 1.25) / (-0.1)

We have two possible solutions for x:

x = (-0.75 + 1.25) / (-0.1) = 0.5 / (-0.1) = -5

x = (-0.75 - 1.25) / (-0.1) = -2 / (-0.1) = 20

Since the value of x cannot be negative, we discard the solution x = -5. Therefore, the ball landed 20 feet away from Jackie.

Conclusion

In this article, we used the equation of a parabola to model the path of a football thrown by Jackie. We then solved for the landing point of the ball by setting the height above the ground to zero and using the quadratic formula. The result showed that the ball landed 20 feet away from Jackie.

Real-World Applications

The concept of parabolas and quadratic equations has numerous real-world applications, including:

  • Projectile motion: The trajectory of a projectile, such as a thrown ball or a rocket, can be modeled using parabolas.
  • Optics: The shape of a mirror or a lens can be described using parabolas.
  • Engineering: Parabolas are used in the design of bridges, arches, and other structures.
  • Physics: The motion of objects under the influence of gravity can be described using parabolas.

Tips and Tricks

  • Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. Make sure to use it correctly to avoid errors.
  • Check your solutions: Always check your solutions to ensure that they make sense in the context of the problem.
  • Use graphing tools: Graphing tools, such as graphing calculators or software, can be useful for visualizing the graph of a parabola and checking your solutions.

Common Mistakes

  • Incorrectly applying the quadratic formula: Make sure to use the quadratic formula correctly to avoid errors.
  • Not checking solutions: Always check your solutions to ensure that they make sense in the context of the problem.
  • Not using graphing tools: Graphing tools can be useful for visualizing the graph of a parabola and checking your solutions.

Conclusion

Q&A: Frequently Asked Questions

Q: What is a parabola?

A: A parabola is a quadratic function that can be expressed in the form:

y = ax^2 + bx + c

where a, b, and c are constants. The graph of a parabola is a U-shaped curve that opens upwards or downwards, depending on the value of a.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the value of a. If a is positive, the parabola opens upwards. If a is negative, the parabola opens downwards.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression and solve for x.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

  • Incorrectly applying the quadratic formula
  • Not checking solutions
  • Not using graphing tools

Q: How do I check my solutions?

A: To check your solutions, you need to plug the values of x back into the original equation and verify that the equation is true.

Q: What are some real-world applications of parabolas?

A: Some real-world applications of parabolas include:

  • Projectile motion
  • Optics
  • Engineering
  • Physics

Q: How do I graph a parabola?

A: To graph a parabola, you need to use a graphing tool, such as a graphing calculator or software. You can also use a table of values to plot the points on the graph.

Q: What are some tips and tricks for working with parabolas?

A: Some tips and tricks for working with parabolas include:

  • Using the quadratic formula correctly
  • Checking solutions
  • Using graphing tools
  • Visualizing the graph of the parabola

Q: How do I determine the vertex of a parabola?

A: To determine the vertex of a parabola, you need to use the formula:

x = -b / 2a

Then, plug the value of x back into the original equation to find the corresponding value of y.

Q: What are some common types of parabolas?

A: Some common types of parabolas include:

  • Upward-opening parabolas
  • Downward-opening parabolas
  • Horizontal parabolas
  • Vertical parabolas

Q: How do I determine the axis of symmetry of a parabola?

A: To determine the axis of symmetry of a parabola, you need to use the formula:

x = -b / 2a

Then, draw a vertical line through the point (x, y) to find the axis of symmetry.

Q: What are some real-world examples of parabolas?

A: Some real-world examples of parabolas include:

  • The trajectory of a thrown ball
  • The shape of a mirror or a lens
  • The design of a bridge or an arch
  • The motion of an object under the influence of gravity

Conclusion

In conclusion, the equation of a parabola is a powerful tool for modeling real-world phenomena, such as the motion of objects. By using the quadratic formula and checking our solutions, we can solve problems involving parabolas and gain a deeper understanding of the world around us.