Converting the Function to Standard Form

Oct 13, 2025 by leolavanderia.com.br 119 views

Introduction

In this article, we will explore the height function of a ball thrown into the air. The height of the ball after a certain number of seconds is given by the function ${$ h = -16(x-2)^2 + 72 $}$. We will first convert this function into standard form and then determine the maximum height of the ball.

Converting the Function to Standard Form

The given function is ${$ h = -16(x-2)^2 + 72 $}$. To convert this function into standard form, we need to expand the squared term and simplify the expression.

Expanding the Squared Term

The squared term in the function is ${$ (x-2)^2 $}$. We can expand this term using the formula ${$ (a-b)^2 = a^2 - 2ab + b^2 $}$.

(x-2)^2 = x^2 - 2x(2) + 2^2
= x^2 - 4x + 4

Substituting the Expanded Term into the Function

Now that we have expanded the squared term, we can substitute it into the original function.

h = -16(x-2)^2 + 72
= -16(x^2 - 4x + 4) + 72
= -16x^2 + 64x - 64 + 72
= -16x^2 + 64x + 8

Simplifying the Expression

The expression ${$ -16x^2 + 64x + 8 $}$ is already in standard form. However, we can simplify it further by factoring out the greatest common factor (GCF) of the terms.

-16x^2 + 64x + 8 = -8(2x^2 - 8x - 1)

The Standard Form of the Function

The standard form of the function is ${$ h = -8(2x^2 - 8x - 1) $}$. This is the equation in standard form.

Determining the Maximum Height

To determine the maximum height of the ball, we need to find the vertex of the parabola represented by the function. The vertex of a parabola in the form ${$ y = ax^2 + bx + c $}$ is given by the formula ${$ x = -\frac{b}{2a} $}$.

Finding the x-Coordinate of the Vertex

In the function ${$ h = -8(2x^2 - 8x - 1) $}$, the coefficient of the squared term is ${$ a = -8(2) = -16 $}$ and the coefficient of the linear term is ${$ b = -8(-8) = 64 $}$. Plugging these values into the formula, we get:

x = -\frac{b}{2a} = -\frac{64}{2(-16)} = -\frac{64}{-32} = 2

Finding the y-Coordinate of the Vertex

Now that we have found the x-coordinate of the vertex, we can plug it into the function to find the y-coordinate.

h = -8(2x^2 - 8x - 1)
= -8(2(2)^2 - 8(2) - 1)
= -8(8 - 16 - 1)
= -8(-9)
= 72

The Maximum Height of the Ball

The maximum height of the ball is ${$ 72 $}$ feet.

Conclusion

$In this article, we have explored the height function of a ball thrown into the air. We have converted the function into standard form and determined the maximum height of the ball. The standard form of the function is ${$ h = -8(2x^2 - 8x - 1) \$}$ and the maximum height of the ball is ${$ 72 \$}$ feet.<br/> **A Ball Thrown into the Air: Understanding the Height Function**$

Q&A: Frequently Asked Questions

In this article, we will answer some frequently asked questions about the height function of a ball thrown into the air.

Q: What is the height function of a ball thrown into the air?

A: The height function of a ball thrown into the air is given by the equation ${$ h = -16(x-2)^2 + 72 $}$. This equation represents the height of the ball after a certain number of seconds.

Q: What is the standard form of the height function?

A: The standard form of the height function is ${$ h = -8(2x^2 - 8x - 1) $}$. This is the equation in standard form.

Q: How do I find the maximum height of the ball?

A: To find the maximum height of the ball, you need to find the vertex of the parabola represented by the function. The vertex of a parabola in the form ${$ y = ax^2 + bx + c $}$ is given by the formula ${$ x = -\frac{b}{2a} $}$. Once you have found the x-coordinate of the vertex, you can plug it into the function to find the y-coordinate, which represents the maximum height of the ball.

Q: What is the maximum height of the ball?

A: The maximum height of the ball is ${$ 72 $}$ feet.

Q: How do I use the height function to determine the height of the ball at a given time?

A: To use the height function to determine the height of the ball at a given time, you need to plug the time into the function. For example, if you want to find the height of the ball after 3 seconds, you would plug x = 3 into the function.

Q: What is the significance of the vertex of the parabola?

A: The vertex of the parabola represents the maximum or minimum point of the function. In this case, the vertex represents the maximum height of the ball.

Q: How do I graph the height function?

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

Q: What are some real-world applications of the height function?

A: The height function has many real-world applications, such as:

Modeling the trajectory of a projectile
Determining the height of a building or a bridge
Calculating the distance between two points
Modeling the motion of an object under the influence of gravity

Conclusion

In this article, we have answered some frequently asked questions about the height function of a ball thrown into the air. We have also provided some real-world applications of the height function.