The Graph Of The Function $f(x)=x^2+8x+12$ Is Shown. Which Statements Describe The Graph? Check All That Apply.- The Vertex Is The Maximum Value.- The Axis Of Symmetry Is $x=-4$.- The Domain Is All Real Numbers.- The Range Is All

Mar 7, 2025 by ADMIN 234 views

**The Graph of a Quadratic Function: Understanding Key Features**

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their graphs is crucial for various applications in science, engineering, and other fields. In this article, we will explore the graph of the function $f(x)=x^2+8x+12$ and identify the key features that describe it.

The Function and Its Graph

The given function is a quadratic function in the form of $f(x)=ax^2+bx+c$ , where $a=1$ , $b=8$ , and $c=12$ . To find the graph of this function, we can start by identifying the vertex, axis of symmetry, domain, and range.

Vertex and Axis of Symmetry

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. To find the vertex, we can use the formula $x=-\frac{b}{2a}$ , where $a$ and $b$ are the coefficients of the quadratic function.

import numpy as np
a = 1
b = 8

x_vertex = -b / (2 * a)
print("The x-coordinate of the vertex is:", x_vertex)

In this case, the x-coordinate of the vertex is $x=-4$ . To find the y-coordinate of the vertex, we can substitute this value into the function.

# Define the function
def f(x):
    return x**2 + 8*x + 12

y_vertex = f(x_vertex)
print("The y-coordinate of the vertex is:", y_vertex)

The y-coordinate of the vertex is $y=20$ . Therefore, the vertex of the graph is the point $(-4, 20)$ .

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex and divides the graph into two symmetrical parts. In this case, the axis of symmetry is the line $x=-4$ .

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 all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Conclusion

Introduction

In our previous article, we explored the graph of the function $f(x)=x^2+8x+12$ and identified its key features, including the vertex, axis of symmetry, domain, and range. In this article, we will answer some frequently asked questions about the graph of a quadratic function.

Q&A

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. It is the minimum or maximum point of the graph, depending on the direction of the parabola.

Q: How do I find the vertex of a quadratic function?

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

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex and divides the graph into two symmetrical parts. It is the line that is equidistant from the two arms of the parabola.

Q: How do I find the axis of symmetry of a quadratic function?

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

Q: What is the domain of a quadratic function?

A: The domain of a quadratic function is the set of all possible input values. For a quadratic function, the domain is all real numbers.

Q: What is the range of a quadratic function?

A: The range of a quadratic function is the set of all possible output values. For a quadratic function, the range is all real numbers greater than or equal to the minimum value of the function.

Q: How do I determine if a quadratic function is a maximum or minimum?

A: To determine if a quadratic function is a maximum or minimum, you can look at the direction of the parabola. If the parabola opens upward, the function is a minimum. If the parabola opens downward, the function is a maximum.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can only have one vertex.

Q: Can a quadratic function have an axis of symmetry that is not a vertical line?

A: No, a quadratic function can only have an axis of symmetry that is a vertical line.

Q: Can a quadratic function have a domain that is not all real numbers?

A: No, a quadratic function can only have a domain that is all real numbers.

Q: Can a quadratic function have a range that is not all real numbers greater than or equal to the minimum value of the function?

A: No, a quadratic function can only have a range that is all real numbers greater than or equal to the minimum value of the function.

Conclusion

In conclusion, the graph of a quadratic function has several key features, including the vertex, axis of symmetry, domain, and range. Understanding these features is crucial for working with quadratic functions and can help you to solve a wide range of problems in mathematics and other fields.