MHF4U Unit I - Polynomial Functions Application [6 Marks]2. A Coin Is Thrown Off A Cliff Into The Lake Below Such That Its Height \[$ H \$\], In Metres, Can Be Modelled By The Function \[$ H(t) = 70 + 6t - 4t^2 \$\], Where Time

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Introduction

In this article, we will explore the application of polynomial functions in a real-world scenario. We will use the concept of polynomial functions to model the height of a coin thrown off a cliff into a lake. The height of the coin can be represented by the function h(t) = 70 + 6t - 4t^2, where t is the time in seconds and h is the height in meters.

Understanding the Problem

The problem involves a coin being thrown off a cliff into a lake. The height of the coin can be modelled by the function h(t) = 70 + 6t - 4t^2, where t is the time in seconds and h is the height in meters. We need to use this function to determine the height of the coin at different times.

Analyzing the Function

The given function h(t) = 70 + 6t - 4t^2 is a quadratic function, which is a type of polynomial function. The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the function is h(t) = 70 + 6t - 4t^2, where a = -4, b = 6, and c = 70.

Graphing the Function

To visualize the function, we can graph it on a coordinate plane. The graph of a quadratic function is a parabola, which is a U-shaped curve. The vertex of the parabola represents the maximum or minimum value of the function.

Finding the Vertex

The vertex of a quadratic function can be found using the formula x = -b/2a. In this case, a = -4 and b = 6, so x = -6/(2(-4)) = 0.75. This means that the vertex of the parabola is at x = 0.75.

Finding the Maximum Height

To find the maximum height of the coin, we need to find the value of h(t) at the vertex of the parabola. We can do this by substituting x = 0.75 into the function h(t) = 70 + 6t - 4t^2. This gives us h(0.75) = 70 + 6(0.75) - 4(0.75)^2 = 70 + 4.5 - 2.25 = 72.25.

Finding the Time of Maximum Height

To find the time at which the coin reaches its maximum height, we need to find the value of t that corresponds to the vertex of the parabola. We can do this by setting x = 0.75 and solving for t. This gives us t = 0.75.

Finding the Height at Different Times

We can use the function h(t) = 70 + 6t - 4t^2 to find the height of the coin at different times. For example, if we want to find the height of the coin at t = 1 second, we can substitute t = 1 into the function. This gives us h(1) = 70 + 6(1) - 4(1)^2 = 70 + 6 - 4 = 72.

Conclusion

In this article, we used the concept of polynomial functions to model the height of a coin thrown off a cliff into a lake. We analyzed the function h(t) = 70 + 6t - 4t^2 and used it to determine the height of the coin at different times. We found the vertex of the parabola and used it to find the maximum height of the coin. We also used the function to find the height of the coin at different times.

Application of Polynomial Functions

Polynomial functions have many applications in real-world scenarios. Some examples include:

  • Modeling population growth: Polynomial functions can be used to model the growth of a population over time.
  • Modeling the motion of objects: Polynomial functions can be used to model the motion of objects, such as the height of a coin thrown off a cliff.
  • Modeling the spread of diseases: Polynomial functions can be used to model the spread of diseases over time.
  • Modeling the growth of economies: Polynomial functions can be used to model the growth of economies over time.

Real-World Examples

Some real-world examples of polynomial functions include:

  • The height of a thrown object: The height of a thrown object can be modelled by a polynomial function, such as h(t) = 70 + 6t - 4t^2.
  • The growth of a population: The growth of a population can be modelled by a polynomial function, such as P(t) = 1000 + 20t - 2t^2.
  • The spread of a disease: The spread of a disease can be modelled by a polynomial function, such as S(t) = 100 + 10t - 2t^2.

Conclusion

In conclusion, polynomial functions have many applications in real-world scenarios. They can be used to model the growth of populations, the motion of objects, the spread of diseases, and the growth of economies. By understanding polynomial functions, we can better understand the world around us and make more informed decisions.

References

  • "Polynomial Functions" by Math Open Reference
  • "Quadratic Functions" by Khan Academy
  • "Modeling with Quadratic Functions" by Math Is Fun

Glossary

  • Polynomial function: A function that can be written in the form f(x) = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants and n is a positive integer.
  • Quadratic function: A polynomial function of degree two, which is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • Vertex: The point on a parabola that represents the maximum or minimum value of the function.
  • Maximum height: The highest point on a parabola, which represents the maximum value of the function.
  • Time of maximum height: The time at which the coin reaches its maximum height.
    MHF4U Unit I - Polynomial Functions Application Q&A =====================================================

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form f(x) = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants and n is a positive integer.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the parabola that represents the maximum or minimum value of the function.

Q: How do you find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula x = -b/2a, where a and b are the coefficients of the quadratic function.

Q: What is the maximum height of a coin thrown off a cliff?

A: The maximum height of a coin thrown off a cliff can be found by using the function h(t) = 70 + 6t - 4t^2, where t is the time in seconds and h is the height in meters.

Q: How do you find the time of maximum height?

A: To find the time of maximum height, you can set x = 0.75 and solve for t, where x is the x-coordinate of the vertex of the parabola.

Q: What are some real-world examples of polynomial functions?

A: Some real-world examples of polynomial functions include:

  • The height of a thrown object: The height of a thrown object can be modelled by a polynomial function, such as h(t) = 70 + 6t - 4t^2.
  • The growth of a population: The growth of a population can be modelled by a polynomial function, such as P(t) = 1000 + 20t - 2t^2.
  • The spread of a disease: The spread of a disease can be modelled by a polynomial function, such as S(t) = 100 + 10t - 2t^2.

Q: What are some applications of polynomial functions?

A: Some applications of polynomial functions include:

  • Modeling population growth: Polynomial functions can be used to model the growth of a population over time.
  • Modeling the motion of objects: Polynomial functions can be used to model the motion of objects, such as the height of a coin thrown off a cliff.
  • Modeling the spread of diseases: Polynomial functions can be used to model the spread of diseases over time.
  • Modeling the growth of economies: Polynomial functions can be used to model the growth of economies over time.

Q: How do you graph a polynomial function?

A: To graph a polynomial function, you can use a graphing calculator or a computer program to plot the function. You can also use a table of values to plot the function.

Q: What are some common mistakes to avoid when working with polynomial functions?

A: Some common mistakes to avoid when working with polynomial functions include:

  • Not simplifying the function: Make sure to simplify the function before graphing or solving it.
  • Not using the correct formula: Make sure to use the correct formula for the vertex of the parabola.
  • Not checking the domain: Make sure to check the domain of the function before graphing or solving it.

Q: How do you determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you can look at the highest power of the variable in the function. For example, if the function is f(x) = 2x^3 + 3x^2 - 4x + 1, the degree of the function is 3.

Q: What is the difference between a polynomial function and a rational function?

A: A polynomial function is a function that can be written in the form f(x) = ax^n + bx^(n-1) + ... + cx + d, where a, b, c, and d are constants and n is a positive integer. A rational function is a function that can be written in the form f(x) = p(x)/q(x), where p(x) and q(x) are polynomial functions.