Modeling the motion of an object under the influence of gravity

Oct 13, 2025 by windailys.com 48 views

The Table of Ordered Pairs Generated from the Quadratic Function $z$

Understanding the Quadratic Function $z$

The quadratic function $z$ is a mathematical equation that generates a set of ordered pairs based on the input values of $x$ . The table provided shows some of the ordered pairs that are generated from this function. In this article, we will delve into the details of the quadratic function $z$ , its characteristics, and the ordered pairs that are generated from it.

Characteristics of the Quadratic Function $z$

The quadratic function $z$ is a polynomial equation of degree two, which means that it has a squared term in the equation. The general form of a quadratic function is $z = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this case, the quadratic function $z$ can be written as $z = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants that are not explicitly given.

The Table of Ordered Pairs

The table provided shows the ordered pairs generated from the quadratic function $z$ . The table has two columns: $x$ and $z(x)$ . The $x$ column represents the input values, and the $z(x)$ column represents the corresponding output values.

$x$	$z(x)$
-3.5	11.25
-2.5	12.25
-1.5	11.25
0	1
1	0
1.5
2.5

Analyzing the Ordered Pairs

From the table, we can see that the ordered pairs generated from the quadratic function $z$ are:

(-3.5, 11.25)
(-2.5, 12.25)
(-1.5, 11.25)
(0, 1)
(1, 0)
(1.5, )
(2.5, )

We can observe that the output values $z(x)$ are increasing as the input values $x$ increase. This suggests that the quadratic function $z$ is an increasing function.

Graphing the Quadratic Function $z$

To visualize the quadratic function $z$ , we can graph the ordered pairs on a coordinate plane. The graph will show the relationship between the input values $x$ and the output values $z(x)$ .

Properties of the Quadratic Function $z$

The quadratic function $z$ has several properties that are worth noting:

Domain: The domain of the quadratic function $z$ is all real numbers, which means that $x$ can take any value.
Range: The range of the quadratic function $z$ is all real numbers greater than or equal to 0, which means that $z(x)$ can take any value greater than or equal to 0.
Vertex: The vertex of the quadratic function $z$ is the point where the function changes from decreasing to increasing. In this case, the vertex is not explicitly given.

Conclusion

In conclusion, the quadratic function $z$ is a mathematical equation that generates a set of ordered pairs based on the input values of $x$ . The table provided shows some of the ordered pairs that are generated from this function. We have analyzed the characteristics of the quadratic function $z$ , the ordered pairs generated from it, and the properties of the function. We have also graphed the quadratic function $z$ to visualize the relationship between the input values $x$ and the output values $z(x)$ .

Further Discussion

The quadratic function $z$ has several applications in mathematics and other fields. For example, it can be used to model the motion of an object under the influence of gravity. It can also be used to solve problems in physics, engineering, and economics.

References

[1] "Quadratic Functions" by Math Open Reference
[2] "Quadratic Equations" by Khan Academy
[3] "Graphing Quadratic Functions" by Purplemath

Additional Resources

[1] "Quadratic Functions" by Wolfram MathWorld
[2] "Quadratic Equations" by Mathway
[3] "Graphing Quadratic Functions" by IXL

Final Thoughts

In conclusion, the quadratic function $z$ is a mathematical equation that generates a set of ordered pairs based on the input values of $x$ . We have analyzed the characteristics of the quadratic function $z$ , the ordered pairs generated from it, and the properties of the function. We have also graphed the quadratic function $z$ to visualize the relationship between the input values $x$ and the output values $z(x)$ . The quadratic function $z$ has several applications in mathematics and other fields, and it is an important concept to understand in mathematics.
Quadratic Function $z$ Q&A

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the quadratic function $z$ . We will cover topics such as the characteristics of the function, the ordered pairs generated from it, and the properties of the function.

Q: What is the quadratic function $z$ ?

A: The quadratic function $z$ is a mathematical equation that generates a set of ordered pairs based on the input values of $x$ . The general form of a quadratic function is $z = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What are the characteristics of the quadratic function $z$ ?

A: The quadratic function $z$ has several characteristics, including:

Domain: The domain of the quadratic function $z$ is all real numbers, which means that $x$ can take any value.
Range: The range of the quadratic function $z$ is all real numbers greater than or equal to 0, which means that $z(x)$ can take any value greater than or equal to 0.
Vertex: The vertex of the quadratic function $z$ is the point where the function changes from decreasing to increasing. In this case, the vertex is not explicitly given.

Q: What are the ordered pairs generated from the quadratic function $z$ ?

A: The ordered pairs generated from the quadratic function $z$ are:

(-3.5, 11.25)
(-2.5, 12.25)
(-1.5, 11.25)
(0, 1)
(1, 0)
(1.5, )
(2.5, )

Q: How can I graph the quadratic function $z$ ?

A: To graph the quadratic function $z$ , you can use a coordinate plane and plot the ordered pairs on it. The graph will show the relationship between the input values $x$ and the output values $z(x)$ .

Q: What are the properties of the quadratic function $z$ ?

A: The quadratic function $z$ has several properties, including:

Symmetry: The quadratic function $z$ is symmetric about the axis of symmetry, which is the vertical line that passes through the vertex of the function.
Increasing and Decreasing: The quadratic function $z$ is increasing for values of $x$ greater than the axis of symmetry and decreasing for values of $x$ less than the axis of symmetry.
Maximum and Minimum: The quadratic function $z$ has a maximum value at the vertex of the function and a minimum value at the endpoints of the function.

Q: How can I use the quadratic function $z$ in real-world applications?

A: The quadratic function $z$ has several real-world applications, including:

Modeling the motion of an object under the influence of gravity: The quadratic function $z$ can be used to model the motion of an object under the influence of gravity.
Solving problems in physics, engineering, and economics: The quadratic function $z$ can be used to solve problems in physics, engineering, and economics.

Q: What are some common mistakes to avoid when working with the quadratic function $z$ ?

A: Some common mistakes to avoid when working with the quadratic function $z$ include:

Not checking the domain and range of the function: Make sure to check the domain and range of the function before using it.
Not identifying the vertex of the function: Make sure to identify the vertex of the function before using it.
Not using the correct formula for the function: Make sure to use the correct formula for the function.

Conclusion

In conclusion, the quadratic function $z$ is a mathematical equation that generates a set of ordered pairs based on the input values of $x$ . We have answered some of the most frequently asked questions about the quadratic function $z$ , including its characteristics, ordered pairs, properties, and real-world applications. We have also discussed some common mistakes to avoid when working with the quadratic function $z$ .