Which Graph Represents $f(x) = (x+2)^2 - 3$?

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Introduction to Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants, and aa cannot be zero.

In this article, we will focus on graphing quadratic functions, specifically the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3. To understand the graph of this function, we need to analyze its properties and behavior.

Properties of Quadratic Functions

Quadratic functions have several properties that are essential to understand when graphing them. Some of these properties include:

  • Vertex: The vertex of a quadratic function is the point where the function changes from decreasing to increasing or vice versa. It is also the minimum or maximum point of the function, depending on the direction of the parabola.
  • Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the function. It is the line that divides the parabola into two equal parts.
  • Intercepts: The x-intercepts are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis.

Graphing f(x)=(x+2)2−3f(x) = (x+2)^2 - 3

To graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3, we need to analyze its properties and behavior. The function is in the form of a quadratic function, and we can identify its vertex, axis of symmetry, and intercepts.

  • Vertex: The vertex of the function is at the point (−2,−3)(-2, -3), which is the minimum point of the function.
  • Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex, which is x=−2x = -2.
  • Intercepts: The x-intercepts are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis.

Graphing the Function

To graph the function, we can use the following steps:

  1. Plot the Vertex: Plot the vertex of the function at the point (−2,−3)(-2, -3).
  2. Plot the Axis of Symmetry: Plot the axis of symmetry at the vertical line x=−2x = -2.
  3. Plot the Intercepts: Plot the x-intercepts at the points where the function crosses the x-axis, and plot the y-intercept at the point where the function crosses the y-axis.
  4. Plot the Parabola: Plot the parabola by connecting the points plotted in the previous steps.

Graph Representation

The graph of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 is a parabola that opens upward. The vertex of the function is at the point (−2,−3)(-2, -3), and the axis of symmetry is the vertical line x=−2x = -2. The x-intercepts are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis.

Conclusion

In conclusion, graphing quadratic functions is an essential concept in mathematics, and understanding the properties and behavior of these functions is crucial. The graph of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 is a parabola that opens upward, with a vertex at the point (−2,−3)(-2, -3) and an axis of symmetry at the vertical line x=−2x = -2. By analyzing the properties and behavior of the function, we can graph it accurately and understand its behavior.

Final Thoughts

Graphing quadratic functions is a fundamental concept in mathematics, and it has numerous applications in various fields. Understanding the properties and behavior of these functions is essential to graph them accurately and analyze their behavior. By following the steps outlined in this article, you can graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 and understand its behavior.

References

  • Algebra: A comprehensive guide to algebra, including quadratic functions and graphing.
  • Geometry: A comprehensive guide to geometry, including quadratic functions and graphing.
  • Calculus: A comprehensive guide to calculus, including quadratic functions and graphing.

Further Reading

  • Graphing Quadratic Functions: A comprehensive guide to graphing quadratic functions, including the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3.
  • Quadratic Functions: A comprehensive guide to quadratic functions, including their properties and behavior.
  • Mathematics: A comprehensive guide to mathematics, including quadratic functions and graphing.

Introduction

Graphing quadratic functions is an essential concept in mathematics, and understanding the properties and behavior of these functions is crucial. In this article, we will answer some frequently asked questions about graphing quadratic functions, specifically the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3.

Q: What is the vertex of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: The vertex of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 is the point (−2,−3)(-2, -3).

Q: What is the axis of symmetry of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: The axis of symmetry of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 is the vertical line x=−2x = -2.

Q: What are the x-intercepts of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: The x-intercepts of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 are the points where the function crosses the x-axis. To find the x-intercepts, we need to set the function equal to zero and solve for x.

Q: How do I graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: To graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3, we need to follow these steps:

  1. Plot the Vertex: Plot the vertex of the function at the point (−2,−3)(-2, -3).
  2. Plot the Axis of Symmetry: Plot the axis of symmetry at the vertical line x=−2x = -2.
  3. Plot the Intercepts: Plot the x-intercepts at the points where the function crosses the x-axis, and plot the y-intercept at the point where the function crosses the y-axis.
  4. Plot the Parabola: Plot the parabola by connecting the points plotted in the previous steps.

Q: What is the y-intercept of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: The y-intercept of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 is the point where the function crosses the y-axis. To find the y-intercept, we need to set x equal to zero and solve for y.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, we need to look at the coefficient of the squared term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

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 function. It is the line that divides the parabola into two equal parts.

Q: How do I find the x-intercepts of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3?

A: To find the x-intercepts of the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3, we need to set the function equal to zero and solve for x.

Q: What is the relationship between the vertex and the axis of symmetry?

A: The vertex and the axis of symmetry are related in that the axis of symmetry passes through the vertex of the function.

Q: How do I graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 using a graphing calculator?

A: To graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 using a graphing calculator, follow these steps:

  1. Enter the function: Enter the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 into the graphing calculator.
  2. Set the window: Set the window to the desired range of x and y values.
  3. Graph the function: Graph the function using the graphing calculator.

Conclusion

In conclusion, graphing quadratic functions is an essential concept in mathematics, and understanding the properties and behavior of these functions is crucial. By following the steps outlined in this article, you can graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 and understand its behavior.

Final Thoughts

Graphing quadratic functions is a fundamental concept in mathematics, and it has numerous applications in various fields. Understanding the properties and behavior of these functions is essential to graph them accurately and analyze their behavior. By following the steps outlined in this article, you can graph the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3 and understand its behavior.

References

  • Algebra: A comprehensive guide to algebra, including quadratic functions and graphing.
  • Geometry: A comprehensive guide to geometry, including quadratic functions and graphing.
  • Calculus: A comprehensive guide to calculus, including quadratic functions and graphing.

Further Reading

  • Graphing Quadratic Functions: A comprehensive guide to graphing quadratic functions, including the function f(x)=(x+2)2−3f(x) = (x+2)^2 - 3.
  • Quadratic Functions: A comprehensive guide to quadratic functions, including their properties and behavior.
  • Mathematics: A comprehensive guide to mathematics, including quadratic functions and graphing.