Give The Domain And Range Of The Quadratic Function Whose Graph Is Described.The Vertex Is \[$(-1,-2)\$\] And The Parabola Opens Upwards.

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Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will focus on the domain and range of a quadratic function whose graph is described. We will use the vertex form of a quadratic function, which is given by:

f(x) = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by:

f(x) = a(x - h)^2 + k

where (h, k) is the vertex of the parabola. In this case, the vertex is given as (-1, -2). Therefore, we can write the quadratic function as:

f(x) = a(x + 1)^2 - 2

Domain of a Quadratic Function

The domain of a quadratic function is the set of all possible input values (x) for which the function is defined. Since a quadratic function is a polynomial function, it is defined for all real numbers. Therefore, the domain of the quadratic function is:

Domain = (-∞, ∞)

Range of a Quadratic Function

The range of a quadratic function is the set of all possible output values (y) for which the function is defined. Since the parabola opens upwards, the minimum value of the function occurs at the vertex. Therefore, the range of the quadratic function is:

Range = [k, ∞)

where k is the y-coordinate of the vertex. In this case, k = -2. Therefore, the range of the quadratic function is:

Range = [-2, ∞)

Graph of a Quadratic Function

The graph of a quadratic function is a parabola that opens upwards or downwards. Since the parabola opens upwards, the graph of the quadratic function is a concave-up parabola. The vertex of the parabola is (-1, -2), and the parabola opens upwards.

Properties of a Quadratic Function

A quadratic function has several important properties, including:

  • Vertex: The vertex of a quadratic function is the point at which the parabola changes direction. In this case, the vertex is (-1, -2).
  • Axis of Symmetry: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. In this case, the axis of symmetry is x = -1.
  • Intercepts: The x-intercepts of a quadratic function are the points at which the parabola intersects the x-axis. In this case, the x-intercepts are not given.
  • Maximum or Minimum Value: The maximum or minimum value of a quadratic function occurs at the vertex. In this case, the minimum value is -2.

Examples of Quadratic Functions

Here are some examples of quadratic functions:

  • f(x) = x^2 + 3x + 2
  • f(x) = -x^2 + 4x - 5
  • f(x) = 2x^2 - 3x + 1

Conclusion

In conclusion, the domain and range of a quadratic function whose graph is described can be determined using the vertex form of a quadratic function. The domain of a quadratic function is the set of all possible input values (x) for which the function is defined, and the range of a quadratic function is the set of all possible output values (y) for which the function is defined. The graph of a quadratic function is a parabola that opens upwards or downwards, and the vertex of the parabola is the point at which the parabola changes direction.

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Vertex Form of a Quadratic Function" by Purplemath
  • [3] "Domain and Range of a Quadratic Function" by Khan Academy

Further Reading

  • "Quadratic Equations" by Math Is Fun
  • "Graphing Quadratic Functions" by Mathway
  • "Quadratic Functions in Real-World Applications" by Wolfram Alpha
    Quadratic Functions: Q&A ==========================

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will provide answers to some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (x) is two. It is typically written in the form:

f(x) = ax^2 + bx + c

where a, b, and c are constants.

Q: What is the vertex form of a quadratic function?

The vertex form of a quadratic function is given by:

f(x) = a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

Q: What is the axis of symmetry of a quadratic function?

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. It is given by the equation:

x = h

where (h, k) is the vertex of the parabola.

Q: How do I find the x-intercepts of a quadratic function?

To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x. This can be done using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic function.

Q: How do I find the maximum or minimum value of a quadratic function?

To find the maximum or minimum value of a quadratic function, you need to find the vertex of the parabola. The vertex is given by the coordinates (h, k), where h is the x-coordinate and k is the y-coordinate.

Q: What is the difference between a quadratic function and a linear function?

A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. A quadratic function has a parabolic shape, while a linear function has a straight line shape.

Q: Can a quadratic function have more than one x-intercept?

Yes, a quadratic function can have more than one x-intercept. This occurs when the quadratic function has two real roots.

Q: Can a quadratic function have no x-intercepts?

Yes, a quadratic function can have no x-intercepts. This occurs when the quadratic function has no real roots.

Q: How do I graph a quadratic function?

To graph a quadratic function, you need to plot the x-intercepts and the vertex of the parabola. You can then use a ruler or a graphing calculator to draw the parabola.

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

Quadratic functions have many real-world applications, including:

  • Projectile motion: Quadratic functions are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
  • Optimization: Quadratic functions are used to optimize problems, such as finding the maximum or minimum value of a function.
  • Physics: Quadratic functions are used to model the motion of objects, such as the motion of a pendulum or the motion of a spring.
  • Engineering: Quadratic functions are used to design and optimize systems, such as the design of a bridge or the optimization of a manufacturing process.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. We hope that this Q&A article has provided you with a better understanding of quadratic functions and their applications.

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Vertex Form of a Quadratic Function" by Purplemath
  • [3] "Domain and Range of a Quadratic Function" by Khan Academy

Further Reading

  • "Quadratic Equations" by Math Is Fun
  • "Graphing Quadratic Functions" by Mathway
  • "Quadratic Functions in Real-World Applications" by Wolfram Alpha