Here Is A Function: $a(x)=x^2-2x+1$.1. Complete The Table For $a(x$\].2. Plot The Coordinate Points.$\[ \begin{tabular}{|c|c|} \hline $x$ & $a(x) = X^2 - 2x + 1$ \\ \hline -2 & \\ \hline -1 & \\ \hline 0 & \\ \hline 1 & \\ \hline 2

Feb 27, 2025 by ADMIN 232 views

Exploring the Quadratic Function: Completing the Table and Plotting Coordinate Points

1. Introduction to Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. In this article, we will explore the quadratic function $a(x) = x^2 - 2x + 1$ and complete the table for its values at different points. We will also plot the coordinate points to visualize the behavior of the function.

2. Completing the Table for $a(x)$

To complete the table for $a(x)$ , we need to substitute the given values of $x$ into the function and calculate the corresponding values of $a(x)$ . Let's start by substituting $x = -2$ into the function:

$a(-2) = (-2)^2 - 2(-2) + 1$ $a(-2) = 4 + 4 + 1$ $a(-2) = 9$

$x$	$a(x) = x^2 - 2x + 1$
-2	9
-1
0
1
2

Next, let's substitute $x = -1$ into the function:

$a(-1) = (-1)^2 - 2(-1) + 1$ $a(-1) = 1 + 2 + 1$ $a(-1) = 4$

$x$	$a(x) = x^2 - 2x + 1$
-2	9
-1	4
0
1
2

Now, let's substitute $x = 0$ into the function:

$a(0) = (0)^2 - 2(0) + 1$ $a(0) = 0 - 0 + 1$ $a(0) = 1$

$x$	$a(x) = x^2 - 2x + 1$
-2	9
-1	4
0	1
1
2

Next, let's substitute $x = 1$ into the function:

$a(1) = (1)^2 - 2(1) + 1$ $a(1) = 1 - 2 + 1$ $a(1) = 0$

$x$	$a(x) = x^2 - 2x + 1$
-2	9
-1	4
0	1
1	0
2

Finally, let's substitute $x = 2$ into the function:

$a(2) = (2)^2 - 2(2) + 1$ $a(2) = 4 - 4 + 1$ $a(2) = 1$

$x$	$a(x) = x^2 - 2x + 1$
-2	9
-1	4
0	1
1	0
2	1

3. Plotting the Coordinate Points

To plot the coordinate points, we need to identify the x-coordinates and the corresponding y-coordinates. The x-coordinates are the values of $x$ that we used to complete the table, and the y-coordinates are the corresponding values of $a(x)$ .

$x$	$a(x)$
-2	9
-1	4
0	1
1	0
2	1

To plot the points, we can use a coordinate plane with the x-axis and y-axis. We can plot each point by marking the corresponding x-coordinate on the x-axis and the corresponding y-coordinate on the y-axis.

4. Conclusion

In this article, we explored the quadratic function $a(x) = x^2 - 2x + 1$ and completed the table for its values at different points. We also plotted the coordinate points to visualize the behavior of the function. The table and the plot provide valuable insights into the behavior of the function, and they can be used to make predictions about the function's behavior at different points.

5. Discussion

The quadratic function $a(x) = x^2 - 2x + 1$ is a simple example of a quadratic function, but it has many interesting properties. For example, the function has a minimum value at $x = 1$ , and the function is increasing for $x > 1$ and decreasing for $x < 1$ . The function also has a symmetry axis at $x = 1$ , which means that the function is symmetric about the line $x = 1$ .

The quadratic function $a(x) = x^2 - 2x + 1$ can be used to model many real-world phenomena, such as the motion of an object under the influence of gravity or the growth of a population over time. The function can also be used to solve problems in physics, engineering, and economics.

In conclusion, the quadratic function $a(x) = x^2 - 2x + 1$ is a fundamental concept in mathematics, and it has many interesting properties and applications. By exploring the function and its behavior, we can gain a deeper understanding of the world around us and develop new insights into the behavior of complex systems.
Quadratic Function Q&A: Exploring the Function and Its Behavior

Introduction

In our previous article, we explored the quadratic function $a(x) = x^2 - 2x + 1$ and completed the table for its values at different points. We also plotted the coordinate points to visualize the behavior of the function. In this article, we will answer some frequently asked questions about the quadratic function and its behavior.

Q: What is the domain of the quadratic function?

A: The domain of the quadratic function is all real numbers, which means that the function is defined for any value of $x$ .

Q: What is the range of the quadratic function?

A: The range of the quadratic function is all real numbers greater than or equal to the minimum value of the function, which is $a(1) = 0$ . This means that the function can take on any value greater than or equal to 0.

Q: What is the vertex of the quadratic function?

A: The vertex of the quadratic function is the point where the function has its minimum value. In this case, the vertex is at $x = 1$ and $y = 0$ .

Q: Is the quadratic function increasing or decreasing?

A: The quadratic function is increasing for $x > 1$ and decreasing for $x < 1$ . This means that the function is increasing as $x$ increases beyond 1 and decreasing as $x$ decreases below 1.

Q: Is the quadratic function symmetric about any axis?

A: Yes, the quadratic function is symmetric about the line $x = 1$ . This means that the function is the same on both sides of the line $x = 1$ .

Q: Can the quadratic function be used to model real-world phenomena?

A: Yes, the quadratic function can be used to model many real-world phenomena, such as the motion of an object under the influence of gravity or the growth of a population over time.

Q: How can the quadratic function be used in physics, engineering, and economics?

A: The quadratic function can be used to solve problems in physics, engineering, and economics. For example, it can be used to model the motion of an object under the influence of gravity, the growth of a population over time, or the cost of producing a product.

Q: Can the quadratic function be used to solve optimization problems?

A: Yes, the quadratic function can be used to solve optimization problems. For example, it can be used to find the maximum or minimum value of a function subject to certain constraints.

Q: How can the quadratic function be used in machine learning and data analysis?

A: The quadratic function can be used in machine learning and data analysis to model complex relationships between variables. For example, it can be used to model the relationship between the price of a product and its demand.

Conclusion

In this article, we answered some frequently asked questions about the quadratic function and its behavior. We hope that this article has provided valuable insights into the properties and applications of the quadratic function. Whether you are a student, a researcher, or a practitioner, the quadratic function is an essential tool that can be used to solve a wide range of problems in mathematics, science, and engineering.

Additional Resources

For more information on the quadratic function and its applications, we recommend the following resources:

We hope that this article has been helpful in your understanding of the quadratic function and its behavior. If you have any further questions or comments, please don't hesitate to contact us.