Graph The Parabola. Provide The Vertex, Axis Of Symmetry, Domain, And Range For The Function:$\[ F(x) = (x-1)^2 - 4 \\]

Introduction

In mathematics, a parabola is a quadratic function that can be represented in various forms, including the vertex form, standard form, and factored form. The vertex form of a parabola is given by the equation $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this article, we will graph the parabola represented by the function $f(x) = (x-1)^2 - 4$ and determine its vertex, axis of symmetry, domain, and range.

Vertex Form of a Parabola

The vertex form of a parabola is given by the equation $f(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. The vertex form is useful for graphing parabolas because it allows us to easily identify the vertex and the direction of the parabola's opening.

Graphing the Parabola

To graph the parabola represented by the function $f(x) = (x-1)^2 - 4$, we can start by identifying the vertex. In this case, the vertex is $(1,-4)$, which is the point where the parabola reaches its minimum or maximum value.

Vertex

The vertex of the parabola is the point where the parabola reaches its minimum or maximum value. In this case, the vertex is $(1,-4)$.

Axis of Symmetry

The axis of symmetry of a parabola 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

The domain of a parabola is the set of all possible input values for which the function is defined. In this case, the domain of the parabola is all real numbers, or $(-\infty,\infty)$.

Range

The range of a parabola is the set of all possible output values for which the function is defined. In this case, the range of the parabola is all real numbers greater than or equal to $-4$, or $[-4,\infty)$.

Determining the Vertex, Axis of Symmetry, Domain, and Range

To determine the vertex, axis of symmetry, domain, and range of the parabola, we can use the following steps:

1. Identify the vertex of the parabola.
2. Identify the axis of symmetry of the parabola.
3. Determine the domain of the parabola.
4. Determine the range of the parabola.

Step 1: Identify the Vertex

The vertex of the parabola is the point where the parabola reaches its minimum or maximum value. In this case, the vertex is $(1,-4)$.

Step 2: Identify the Axis of Symmetry

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

Step 3: Determine the Domain

The domain of a parabola is the set of all possible input values for which the function is defined. In this case, the domain of the parabola is all real numbers, or $(-\infty,\infty)$.

Step 4: Determine the Range

The range of a parabola is the set of all possible output values for which the function is defined. In this case, the range of the parabola is all real numbers greater than or equal to $-4$, or $[-4,\infty)$.

Conclusion

In this article, we graphed the parabola represented by the function $f(x) = (x-1)^2 - 4$ and determined its vertex, axis of symmetry, domain, and range. The vertex of the parabola is $(1,-4)$, the axis of symmetry is the line $x=1$, the domain is all real numbers, and the range is all real numbers greater than or equal to $-4$. By following the steps outlined in this article, you can determine the vertex, axis of symmetry, domain, and range of any parabola.

Example Problems

Problem 1

Graph the parabola represented by the function $f(x) = (x-2)^2 - 3$ and determine its vertex, axis of symmetry, domain, and range.

Solution

The vertex of the parabola is $(2,-3)$, the axis of symmetry is the line $x=2$, the domain is all real numbers, and the range is all real numbers greater than or equal to $-3$.

Problem 2

Graph the parabola represented by the function $f(x) = (x+1)^2 + 2$ and determine its vertex, axis of symmetry, domain, and range.

Solution

The vertex of the parabola is $(-1,2)$, the axis of symmetry is the line $x=-1$, the domain is all real numbers, and the range is all real numbers greater than or equal to $2$.

Final Thoughts

Graphing parabolas and determining their vertex, axis of symmetry, domain, and range is an important skill in mathematics. By following the steps outlined in this article, you can graph any parabola and determine its key characteristics. Whether you are a student or a teacher, this skill is essential for understanding and working with quadratic functions.

Introduction

In our previous article, we graphed the parabola represented by the function $f(x) = (x-1)^2 - 4$ and determined its vertex, axis of symmetry, domain, and range. In this article, we will answer some frequently asked questions about graphing parabolas and determining their key characteristics.

Q&A

Q1: What is the vertex of a parabola?

A1: The vertex of a parabola is the point where the parabola reaches its minimum or maximum value. It is represented by the point $(h,k)$ in the vertex form of a parabola, $f(x) = a(x-h)^2 + k$.

Q2: How do I determine the axis of symmetry of a parabola?

A2: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. To determine the axis of symmetry, you can use the formula $x = h$, where $(h,k)$ is the vertex of the parabola.

Q3: What is the domain of a parabola?

A3: The domain of a parabola is the set of all possible input values for which the function is defined. For a parabola in the form $f(x) = a(x-h)^2 + k$, the domain is all real numbers, or $(-\infty,\infty)$.

Q4: What is the range of a parabola?

A4: The range of a parabola is the set of all possible output values for which the function is defined. For a parabola in the form $f(x) = a(x-h)^2 + k$, the range is all real numbers greater than or equal to $k$, or $[k,\infty)$.

Q5: How do I graph a parabola?

A5: To graph a parabola, you can use the following steps:

1. Identify the vertex of the parabola.
2. Identify the axis of symmetry of the parabola.
3. Plot the vertex and the axis of symmetry on a coordinate plane.
4. Use the vertex and the axis of symmetry to determine the direction of the parabola's opening.
5. Plot additional points on the parabola to complete the graph.

Q6: What is the difference between a parabola and a quadratic function?

A6: A parabola is a graphical representation of a quadratic function, while a quadratic function is a mathematical expression that can be represented graphically as a parabola. In other words, a parabola is a visual representation of a quadratic function, while a quadratic function is a mathematical expression that can be graphed as a parabola.

Q7: Can a parabola have a negative vertex?

A7: Yes, a parabola can have a negative vertex. In fact, the vertex of a parabola can be any point on the coordinate plane, including points with negative coordinates.

Q8: How do I determine the equation of a parabola?

A8: To determine the equation of a parabola, you can use the following steps:

1. Identify the vertex of the parabola.
2. Identify the axis of symmetry of the parabola.
3. Use the vertex and the axis of symmetry to determine the direction of the parabola's opening.
4. Use the vertex and the axis of symmetry to determine the equation of the parabola in the form $f(x) = a(x-h)^2 + k$.

Conclusion

In this article, we answered some frequently asked questions about graphing parabolas and determining their key characteristics. We hope that this article has been helpful in clarifying any confusion you may have had about graphing parabolas and determining their vertex, axis of symmetry, domain, and range.

Example Problems

Problem 1

Graph the parabola represented by the function $f(x) = (x-2)^2 - 3$ and determine its vertex, axis of symmetry, domain, and range.

Solution

The vertex of the parabola is $(2,-3)$, the axis of symmetry is the line $x=2$, the domain is all real numbers, and the range is all real numbers greater than or equal to $-3$.

Problem 2

Graph the parabola represented by the function $f(x) = (x+1)^2 + 2$ and determine its vertex, axis of symmetry, domain, and range.

Solution

The vertex of the parabola is $(-1,2)$, the axis of symmetry is the line $x=-1$, the domain is all real numbers, and the range is all real numbers greater than or equal to $2$.

Final Thoughts

Graphing parabolas and determining their key characteristics is an important skill in mathematics. By following the steps outlined in this article, you can graph any parabola and determine its vertex, axis of symmetry, domain, and range. Whether you are a student or a teacher, this skill is essential for understanding and working with quadratic functions.