3. \[$ Y = X^2 - 2x \$\]- Increase:- Decrease:- Axis Of Symmetry: - Vertex: - Domain: - Range: $\[ \begin{tabular}{|c|c|} \hline x & Y \\ \hline & \\ \hline & \\ \hline & \\ \hline & \\ \hline &

Understanding Quadratic Functions: A Comprehensive Analysis of $y = x^2 - 2x$

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will delve into the analysis of the quadratic function $y = x^2 - 2x$, exploring its key characteristics, including increase, decrease, axis of symmetry, vertex, domain, and range.

Increase and Decrease

To determine the intervals where the function $y = x^2 - 2x$ is increasing or decreasing, we need to find the critical points. The critical points occur when the derivative of the function is equal to zero or undefined. Let's find the derivative of the function:

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

Now, we set the derivative equal to zero and solve for $x$:

$$2x - 2 = 0$$

$$2x = 2$$

$$x = 1$$

The critical point is $x = 1$. To determine the intervals where the function is increasing or decreasing, we can use the first derivative test. We choose a point to the left of the critical point and a point to the right of the critical point. Let's choose $x = 0$ and $x = 2$.

For $x = 0$, we have:

$$\frac{dy}{dx} = 2(0) - 2 = -2$$

Since the derivative is negative, the function is decreasing at $x = 0$.

For $x = 2$, we have:

$$\frac{dy}{dx} = 2(2) - 2 = 2$$

Since the derivative is positive, the function is increasing at $x = 2$.

Now, we can conclude that the function $y = x^2 - 2x$ is decreasing on the interval $(-\infty, 1)$ and increasing on the interval $(1, \infty)$.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. To find the axis of symmetry, we need to find the x-coordinate of the vertex. The x-coordinate of the vertex is given by:

$$x = -\frac{b}{2a}$$

In this case, $a = 1$ and $b = -2$. Plugging in these values, we get:

$$x = -\frac{-2}{2(1)} = 1$$

So, the axis of symmetry is the vertical line $x = 1$.

Vertex

The vertex is the point on the parabola where the function changes from decreasing to increasing or vice versa. We have already found the x-coordinate of the vertex, which is $x = 1$. To find the y-coordinate of the vertex, we plug in $x = 1$ into the original function:

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

So, the vertex 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 this case, the function $y = x^2 - 2x$ is defined for all real numbers, so the domain is:

$$(-\infty, \infty)$$

Range

The range of a function is the set of all possible output values for which the function is defined. To find the range, we need to find the minimum and maximum values of the function. We have already found the vertex, which is the minimum point on the parabola. The minimum value of the function is $-1$, which occurs at $x = 1$. To find the maximum value, we can plug in a large positive value of $x$ into the original function. Let's choose $x = 100$:

$$y = (100)^2 - 2(100) = 10000 - 200 = 9800$$

So, the maximum value of the function is $9800$, which occurs at $x = 100$. Therefore, the range of the function is:

$$(-\infty, 9800]$$

Conclusion

In this article, we have analyzed the quadratic function $y = x^2 - 2x$, exploring its key characteristics, including increase, decrease, axis of symmetry, vertex, domain, and range. We have found that the function is decreasing on the interval $(-\infty, 1)$ and increasing on the interval $(1, \infty)$. The axis of symmetry is the vertical line $x = 1$, and the vertex is the point $(1, -1)$. The domain of the function is $(-\infty, \infty)$, and the range is $(-\infty, 9800]$.
Quadratic Function $y = x^2 - 2x$: A Comprehensive Q&A Guide

In our previous article, we explored the key characteristics of the quadratic function $y = x^2 - 2x$, including increase, decrease, axis of symmetry, vertex, domain, and range. In this article, we will provide a Q&A guide to help you better understand and work with this function.

Q: What is the equation of the axis of symmetry?

A: The equation of the axis of symmetry is $x = 1$.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point $(1, -1)$.

Q: What is the domain of the function?

A: The domain of the function is $(-\infty, \infty)$.

Q: What is the range of the function?

A: The range of the function is $(-\infty, 9800]$.

Q: Is the function increasing or decreasing on the interval $(-\infty, 1)$?

A: The function is decreasing on the interval $(-\infty, 1)$.

Q: Is the function increasing or decreasing on the interval $(1, \infty)$?

A: The function is increasing on the interval $(1, \infty)$.

Q: How do I find the x-coordinate of the vertex?

A: To find the x-coordinate of the vertex, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you can plug in the x-coordinate of the vertex into the original function.

Q: What is the significance of the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is a line of symmetry, meaning that the parabola is reflected about this line.

Q: How do I graph the function?

A: To graph the function, you can start by plotting the vertex and the axis of symmetry. Then, you can use the information about the increase and decrease of the function to plot additional points.

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
• Finding the maximum or minimum value of a function
• Solving optimization problems
• Modeling population growth or decline
• Finding the area or perimeter of a shape

Conclusion

In this Q&A guide, we have provided answers to common questions about the quadratic function $y = x^2 - 2x$. We hope that this guide has been helpful in understanding and working with this function. If you have any further questions, please don't hesitate to ask.

Additional Resources

• For more information about quadratic functions, please see our previous article on the topic.
• For practice problems and exercises, please see the following resources:
  • Khan Academy: Quadratic Functions
  • Mathway: Quadratic Functions
  • Wolfram Alpha: Quadratic Functions

Final Thoughts

Quadratic functions are a fundamental concept in mathematics, and they have many real-world applications. By understanding and working with quadratic functions, you can develop problem-solving skills and apply mathematical concepts to real-world problems. We hope that this Q&A guide has been helpful in your journey to learn about quadratic functions.