Graph The Parabola. Provide The Vertex, Axis Of Symmetry, Domain, And Range.$\[ F(x) = 2(x-1)^2 - 2 \\] Use The Graphing Tool To Graph The Parabola.

Introduction

In mathematics, a parabola is a fundamental concept that represents a quadratic function. It is a U-shaped curve that can be graphed using various methods, including algebraic and graphical techniques. In this article, we will explore the process of graphing a parabola, identifying its key features, and understanding its properties. We will use the given function, ${ f(x) = 2(x-1)^2 - 2 }$, to demonstrate the graphing process and analyze its characteristics.

Understanding the Function

Before we begin graphing the parabola, let's take a closer look at the given function:

${ f(x) = 2(x-1)^2 - 2 }$

This function is a quadratic function in the form of ${ f(x) = a(x-h)^2 + k }$, where:

• ${ a = 2 }$ is the coefficient of the squared term
• ${ h = 1 }$ is the x-coordinate of the vertex
• ${ k = -2 }$ is the y-coordinate of the vertex

Graphing the Parabola

To graph the parabola, we can use a graphing tool or software. However, we can also use algebraic techniques to identify the key features of the parabola. Let's start by identifying the vertex of the parabola.

Vertex of the Parabola

The vertex of the parabola is the point where the parabola changes direction. In this case, the vertex is located at the point ${ (h, k) = (1, -2) }$. To find the vertex, we can use the formula:

${ (h, k) = \left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right) }$

However, since the function is already in vertex form, we can simply read the vertex coordinates from the function.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line ${ x = 1 }$.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For a quadratic function, the domain is always all real numbers, while the range depends on the coefficient of the squared term.

In this case, the domain of the function is all real numbers, while the range is all real numbers greater than or equal to -2.

Graphing the Parabola Using a Graphing Tool

Now that we have identified the key features of the parabola, let's use a graphing tool to graph the parabola.

Graphing the Parabola

To graph the parabola, we can use a graphing tool such as Desmos or Graphing Calculator. We can enter the function ${ f(x) = 2(x-1)^2 - 2 }$ into the graphing tool and adjust the settings to display the graph.

Graphing the Parabola

Here is the graph of the parabola:

Graph of the Parabola

The graph of the parabola is a U-shaped curve that opens upward. The vertex of the parabola is located at the point ${ (1, -2) }$, and the axis of symmetry is the line ${ x = 1 }$. The domain of the function is all real numbers, while the range is all real numbers greater than or equal to -2.

Conclusion

In this article, we have graphed the parabola using a given function, identified its key features, and analyzed its properties. We have used algebraic techniques to identify the vertex, axis of symmetry, domain, and range of the parabola. We have also used a graphing tool to graph the parabola and visualize its characteristics.

Key Takeaways

• The vertex of a parabola is the point where the parabola changes direction.
• The axis of symmetry is a vertical line that passes through the vertex of the parabola.
• The domain of a quadratic function is all real numbers, while the range depends on the coefficient of the squared term.
• A graphing tool can be used to graph a parabola and visualize its characteristics.

Final Thoughts

Introduction

In our previous article, we graphed the parabola using a given function, identified its key features, and analyzed its properties. In this article, we will answer some frequently asked questions about graphing parabolas and provide additional insights and knowledge.

Q&A

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on whether the parabola opens upward or downward.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula:

${ (h, k) = \left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right) }$

However, if the function is already in vertex form, you can simply read the vertex coordinates from the function.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola 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 of a parabola?

A: To find the axis of symmetry of a parabola, you can use the formula:

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

However, if the function is already in vertex form, you can simply read the axis of symmetry from the function.

Q: What is the domain and range of a parabola?

A: The domain of a parabola is the set of all possible input values, while the range is the set of all possible output values. For a quadratic function, the domain is always all real numbers, while the range depends on the coefficient of the squared term.

Q: How do I graph a parabola?

A: To graph a parabola, you can use a graphing tool or software. You can also use algebraic techniques to identify the key features of the parabola, such as the vertex, axis of symmetry, domain, and range.

Q: What are some common mistakes to avoid when graphing a parabola?

A: Some common mistakes to avoid when graphing a parabola include:

• Not identifying the vertex and axis of symmetry correctly
• Not using the correct formula to find the vertex and axis of symmetry
• Not considering the domain and range of the parabola
• Not using a graphing tool or software to graph the parabola

Conclusion

In this article, we have answered some frequently asked questions about graphing parabolas and provided additional insights and knowledge. We hope that this article has been helpful in clarifying any doubts or questions you may have had about graphing parabolas.

Key Takeaways

• The vertex of a parabola is the point where the parabola changes direction.
• The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola.
• The domain of a parabola is the set of all possible input values, while the range is the set of all possible output values.
• A graphing tool or software can be used to graph a parabola and visualize its characteristics.

Final Thoughts

Graphing a parabola is an essential skill in mathematics, and it requires a deep understanding of quadratic functions and their properties. By graphing a parabola, we can identify its key features, analyze its characteristics, and visualize its behavior. We hope that this article has provided valuable insights and knowledge to readers.