Given The Function $y = X^2 - 2x$, Determine The Following:- Axis Of Symmetry: $\qquad$- Vertex: $\qquad$- Domain: $\qquad$- Range: $\qquad$Complete The Table Of

Feb 26, 2025 by ADMIN 162 views

**Understanding the Properties of a Quadratic Function**

Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on the function $y = x^2 - 2x$ and determine its axis of symmetry, vertex, domain, and range. We will also complete a table of values for the function.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. To find the axis of symmetry, we need to find the value of $x$ that makes the derivative of the function equal to zero.

The derivative of the function $y = x^2 - 2x$ is given by:

\frac{dy}{dx} = 2x - 2

Setting the derivative equal to zero, we get:

2x - 2 = 0

Solving for $x$ , we get:

x = 1

Therefore, the axis of symmetry of the function $y = x^2 - 2x$ is the vertical line $x = 1$ .

Vertex

The vertex of a quadratic function is the point on the parabola where the function changes from decreasing to increasing or vice versa. The vertex is also the minimum or maximum point of the parabola.

To find the vertex, we need to find the value of $x$ that makes the derivative of the function equal to zero. We have already done this in the previous section, and we found that the axis of symmetry is the vertical line $x = 1$ .

The vertex of the parabola is the point on the axis of symmetry that is closest to the origin. To find the vertex, we need to substitute the value of $x$ into the original function.

Substituting $x = 1$ into the function $y = x^2 - 2x$ , we get:

y = (1)^2 - 2(1)

Simplifying, we get:

y = 1 - 2

y = -1

Therefore, the vertex of the parabola is the point $(1, -1)$ .

Domain

The domain of a function is the set of all possible input values for which the function is defined. In the case of a quadratic function, the domain is all real numbers.

However, if the function is not defined for certain values of $x$ , then those values are not included in the domain. In this case, the function is not defined for $x = 0$ because it would result in division by zero.

Therefore, the domain of the function $y = x^2 - 2x$ is all real numbers except $x = 0$ .

Range

The range of a function is the set of all possible output values for which the function is defined. In the case of a quadratic function, the range is all real numbers.

However, if the function has a minimum or maximum value, then the range is limited to the values between the minimum and maximum. In this case, the function has a minimum value of $-1$ at the vertex.

Therefore, the range of the function $y = x^2 - 2x$ is all real numbers greater than or equal to $-1$ .

Table of Values

To complete the table of values, we need to find the values of $y$ for different values of $x$ . We can do this by substituting the values of $x$ into the original function.

$x$	$y$
-2	0
-1	1
0	0
1	-1
2	0
3	5

Conclusion

In this article, we have determined the axis of symmetry, vertex, domain, and range of the function $y = x^2 - 2x$ . We have also completed a table of values for the function.

The axis of symmetry is the vertical line $x = 1$ , the vertex is the point $(1, -1)$ , the domain is all real numbers except $x = 0$ , and the range is all real numbers greater than or equal to $-1$ .

Introduction

In our previous article, we discussed the properties of a quadratic function, including the axis of symmetry, vertex, domain, and range. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. It is typically written in the form $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is the line that divides the parabola into two equal parts.

Q: How do I find the axis of symmetry?

A: To find the axis of symmetry, you need to find the value of $x$ that makes the derivative of the function equal to zero. This value is the $x$ -coordinate of the vertex.

Q: What is the vertex?

A: The vertex is the point on the parabola where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the parabola.

Q: How do I find the vertex?

A: To find the vertex, you need to find the value of $x$ that makes the derivative of the function equal to zero. This value is the $x$ -coordinate of the vertex. You can then substitute this value into the original function to find the $y$ -coordinate of the vertex.

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is all real numbers, unless the function is not defined for certain values of $x$ . In this case, those values are not included in the domain.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is all real numbers, unless the function has a minimum or maximum value. In this case, the range is limited to the values between the minimum and maximum.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the points on the graph and then draw a smooth curve through them. You can also use a graphing calculator or software to graph the function.

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

A: Quadratic functions have numerous real-world applications, including:

Physics: Quadratic functions are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic functions are used to design bridges, buildings, and other structures.
Economics: Quadratic functions are used to model the behavior of economic systems.
Computer Science: Quadratic functions are used in algorithms and data structures.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We hope that this article has provided a clear understanding of the properties and applications of quadratic functions.

Additional Resources

For more information on quadratic functions, we recommend the following resources:

Khan Academy: Quadratic Functions
Mathway: Quadratic Functions
Wolfram Alpha: Quadratic Functions

We hope that this article has been helpful in your understanding of quadratic functions. If you have any further questions, please don't hesitate to ask.