The Table Shows The Approximate Height Of An Object $x$ Seconds After The Object Was Dropped. The Function H ( X ) = − 16 X 2 + 100 H(x) = -16x^2 + 100 H ( X ) = − 16 X 2 + 100 Models The Data In The Table.Height Of A Dropped Object$[ \begin{tabular}{|c|c|} \hline \text{Time

Modeling the Height of a Dropped Object Using a Quadratic Function

The table shows the approximate height of an object $x$ seconds after the object was dropped. The function $h(x) = -16x^2 + 100$ models the data in the table. This function is a quadratic function, which is a polynomial function of degree two. In this article, we will explore the properties of this function and how it models the height of the object.

Understanding the Quadratic Function

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In the function $h(x) = -16x^2 + 100$, the coefficient of the squared term is $-16$, the coefficient of the linear term is $0$, and the constant term is $100$.

Properties of the Quadratic Function

The function $h(x) = -16x^2 + 100$ has several properties that are important to understand. First, the coefficient of the squared term, $-16$, is negative, which means the function is a downward-opening parabola. This means that the function will have a maximum value, which occurs at the vertex of the parabola.

Finding the Vertex of the Parabola

The vertex of a parabola is the point at which the function changes from increasing to decreasing or vice versa. To find the vertex of the parabola, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = -16$ and $b = 0$, so the x-coordinate of the vertex is $x = -\frac{0}{2(-16)} = 0$. To find the y-coordinate of the vertex, we can plug this value into the function: $h(0) = -16(0)^2 + 100 = 100$. Therefore, the vertex of the parabola is at the point $(0, 100)$.

Understanding the Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is $x = -\frac{b}{2a}$. In this case, the equation of the axis of symmetry is $x = 0$. This means that the axis of symmetry is the y-axis.

Understanding the Intercepts

The x-intercepts of a parabola are the points at which the function crosses the x-axis. To find the x-intercepts, we can set the function equal to zero and solve for $x$. In this case, we have $-16x^2 + 100 = 0$. We can solve this equation by factoring: $-16x^2 = -100$, $x^2 = \frac{100}{16}$, $x^2 = \frac{25}{4}$, $x = \pm\frac{5}{2}$. Therefore, the x-intercepts are at the points $(-\frac{5}{2}, 0)$ and $(\frac{5}{2}, 0)$.

Understanding the Domain and Range

The domain of a function is the set of all input values. The range of a function is the set of all possible output values. In this case, the domain of the function is all real numbers, since we can plug in any value of $x$ into the function. The range of the function is all real numbers greater than or equal to $0$, since the function will never be negative.

Conclusion

In this article, we have explored the properties of the quadratic function $h(x) = -16x^2 + 100$. We have found the vertex of the parabola, the axis of symmetry, the x-intercepts, and the domain and range of the function. This function models the height of an object that is dropped from a certain height. The function is a downward-opening parabola, which means it will have a maximum value at the vertex of the parabola. The function will never be negative, since the range of the function is all real numbers greater than or equal to $0$.

Applications of the Quadratic Function

The quadratic function $h(x) = -16x^2 + 100$ has several applications in real-world situations. For example, it can be used to model the height of an object that is dropped from a certain height. It can also be used to model the trajectory of a projectile, such as a ball or a rocket. Additionally, it can be used to model the growth of a population over time.

Real-World Examples

There are several real-world examples of the quadratic function $h(x) = -16x^2 + 100$. For example, it can be used to model the height of a building or a bridge. It can also be used to model the trajectory of a ball or a rocket. Additionally, it can be used to model the growth of a population over time.

Conclusion

In conclusion, the quadratic function $h(x) = -16x^2 + 100$ is a powerful tool for modeling real-world situations. It can be used to model the height of an object that is dropped from a certain height, the trajectory of a projectile, and the growth of a population over time. Its properties, such as the vertex of the parabola, the axis of symmetry, the x-intercepts, and the domain and range, make it a useful tool for understanding and analyzing real-world data.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Quadratic Equations" by Math Is Fun
• [3] "Quadratic Functions and Equations" by Khan Academy

Further Reading

• "Quadratic Functions and Equations" by Khan Academy
• "Quadratic Functions" by Math Open Reference
• "Quadratic Equations" by Math Is Fun

Final Thoughts

In conclusion, the quadratic function $h(x) = -16x^2 + 100$ is a powerful tool for modeling real-world situations. Its properties, such as the vertex of the parabola, the axis of symmetry, the x-intercepts, and the domain and range, make it a useful tool for understanding and analyzing real-world data.

Modeling the Height of a Dropped Object Using a Quadratic Function

Q&A: Quadratic Functions and the Height of a Dropped Object

In this article, we will continue to explore the properties of the quadratic function $h(x) = -16x^2 + 100$ and answer some common questions about quadratic functions and the height of a dropped object.

Q: What is a quadratic function?

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

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the function changes from increasing to decreasing or vice versa. To find the vertex of a parabola, we can use the formula $x = -\frac{b}{2a}$.

Q: What is the axis of symmetry?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The equation of the axis of symmetry is $x = -\frac{b}{2a}$.

Q: What are the x-intercepts of a parabola?

A: The x-intercepts of a parabola are the points at which the function crosses the x-axis. To find the x-intercepts, we can set the function equal to zero and solve for $x$.

Q: What is the domain and range of a quadratic function?

A: The domain of a function is the set of all input values. The range of a function is the set of all possible output values. In the case of a quadratic function, the domain is all real numbers, and the range is all real numbers greater than or equal to $0$.

Q: How can I use a quadratic function to model the height of a dropped object?

A: To use a quadratic function to model the height of a dropped object, you can use the formula $h(x) = -16x^2 + 100$, where $x$ is the time in seconds and $h(x)$ is the height of the object at time $x$.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including modeling the trajectory of a projectile, the growth of a population over time, and the height of an object that is dropped from a certain height.

Q: How can I find the vertex of a parabola if I don't know the equation of the parabola?

A: If you don't know the equation of the parabola, you can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can plug this value into the equation of the parabola.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. This means that a quadratic function has a squared term, while a linear function does not.

Q: How can I use a quadratic function model the growth of a population over time?

A: To use a quadratic function to model the growth of a population over time, you can use the formula $p(t) = at^2 + bt + c$, where $p(t)$ is the population at time $t$ and $a$, $b$, and $c$ are constants.

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

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

• Not using the correct formula to find the vertex of a parabola
• Not using the correct formula to find the x-intercepts of a parabola
• Not checking the domain and range of a quadratic function
• Not using the correct formula to model the growth of a population over time

Conclusion

In this article, we have answered some common questions about quadratic functions and the height of a dropped object. We have also explored the properties of the quadratic function $h(x) = -16x^2 + 100$ and provided some real-world applications of quadratic functions.