Graph The Function:$\[ G(x) = -x^2 - 6x - 6 \\]

Introduction

Graphing quadratic functions is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on graphing the function $g(x) = -x^2 - 6x - 6$. We will explore the properties of this function, including its vertex, axis of symmetry, and x-intercepts.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

Graphing the Function $g(x) = -x^2 - 6x - 6$

To graph the function $g(x) = -x^2 - 6x - 6$, we need to identify its vertex, axis of symmetry, and x-intercepts.

Vertex

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 $x = -\frac{b}{2a}$. In this case, $a = -1$ and $b = -6$, so $x = -\frac{-6}{2(-1)} = 3$. To find the y-coordinate of the vertex, we substitute $x = 3$ into the function: $g(3) = -(3)^2 - 6(3) - 6 = -9 - 18 - 6 = -33$. Therefore, the vertex of the graph is $(3, -33)$.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. In this case, the axis of symmetry is the line $x = 3$.

X-Intercepts

The x-intercepts of a quadratic function are the points on the graph where the function intersects the x-axis. To find the x-intercepts, we set $g(x) = 0$ and solve for x. In this case, we have $-x^2 - 6x - 6 = 0$. We can factor the left-hand side of the equation: $-(x + 3)(x + 2) = 0$. Therefore, the x-intercepts are $x = -3$ and $x = -2$.

Graphing the Function

Now that we have identified the vertex, axis of symmetry, and x-intercepts, we can graph the function $g(x) = -x^2 - 6x - 6$. The graph is a parabola that opens downward, with its vertex at $(3, -33)$ and x-intercepts at $x = -3$ and $x = -2$.

Properties of the Graph

The graph of the function $g(x) = -x^2 - 6x - 6$ has several properties that are worth noting:

• Vertex: The vertex of the graph is $(3, -33)$.
• Axis of Symmetry: The axis of symmetry is the line $x = 3$.
• X-Intercepts: The x-intercepts are $x = -3$ and $x = -2$.
• Domain: The domain of the graph is all real numbers.
• Range: The range of the graph is all real numbers less than or equal to -33.

Real-World Applications

The graph of the function $g(x) = -x^2 - 6x - 6$ has several real-world applications:

• Physics: The graph can be used to model the motion of an object under the influence of gravity.
• Engineering: The graph can be used to design the shape of a parabolic reflector or antenna.
• Economics: The graph can be used to model the demand for a product or service.

Conclusion

In conclusion, graphing the function $g(x) = -x^2 - 6x - 6$ requires identifying its vertex, axis of symmetry, and x-intercepts. The graph is a parabola that opens downward, with its vertex at $(3, -33)$ and x-intercepts at $x = -3$ and $x = -2$. The graph has several properties, including a vertex, axis of symmetry, x-intercepts, domain, and range. The graph has several real-world applications, including physics, engineering, and economics.

References

• [1] "Graphing Quadratic Functions". Math Open Reference.
• [2] "Quadratic Functions". Khan Academy.
• [3] "Graphing Quadratic Functions". Purplemath.

Additional Resources

• [1] "Graphing Quadratic Functions" by Math Open Reference.
• [2] "Quadratic Functions" by Khan Academy.
• [3] "Graphing Quadratic Functions" by Purplemath.

Frequently Asked Questions

• Q: What is the vertex of the graph of the function $g(x) = -x^2 - 6x - 6$? A: The vertex of the graph is $(3, -33)$.
• Q: What is the axis of symmetry of the graph of the function $g(x) = -x^2 - 6x - 6$? A: The axis of symmetry is the line $x = 3$.
• Q: What are the x-intercepts of the graph of the function $g(x) = -x^2 - 6x - 6$? A: The x-intercepts are $x = -3$ and $x = -2$.
  Graphing Quadratic Functions: A Comprehensive Guide =====================================================

Q&A: Graphing Quadratic Functions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to identify its vertex, axis of symmetry, and x-intercepts. You can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then substitute this value into the function to find the y-coordinate of the vertex.

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.

Q: How do I find 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. You can find the axis of symmetry by using the formula $x = -\frac{b}{2a}$.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points on the graph where the function intersects the x-axis. You can find the x-intercepts by setting the function equal to zero and solving for x.

Q: How do I graph the function $g(x) = -x^2 - 6x - 6$?

A: To graph the function $g(x) = -x^2 - 6x - 6$, you need to identify its vertex, axis of symmetry, and x-intercepts. The vertex is $(3, -33)$, the axis of symmetry is the line $x = 3$, and the x-intercepts are $x = -3$ and $x = -2$.

Q: What are the properties of the graph of the function $g(x) = -x^2 - 6x - 6$?

A: The graph of the function $g(x) = -x^2 - 6x - 6$ has several properties, including a vertex, axis of symmetry, x-intercepts, domain, and range. The vertex is $(3, -33)$, the axis of symmetry is the line $x = 3$, the x-intercepts are $x = -3$ and $x = -2$, the domain is all real numbers, and the range is all real numbers less than or equal to -33.

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

A: The graph of a quadratic function has several real-world applications, including physics, engineering, and economics. In physics, the graph can be used to model the motion of an object under the influence of gravity. In engineering, the graph can be used to design the shape of a parabolic reflector or antenna. In economics, the graph can be used to model the demand for a product or service.

Q: How do I use graphing quadratic functions in real-world applications?

A: To use graphing quadratic functions in real-world applications, you need to identify the properties of the graph, such as the vertex, axis of symmetry, and x-intercepts. You can then use this information to model real-world phenomena, such as the motion of an object under the influence of gravity or the demand for a product or service.

Q: What are some common mistakes to avoid when graphing quadratic functions?

A: Some common mistakes to avoid when graphing quadratic functions include:

• Not identifying the vertex, axis of symmetry, and x-intercepts of the graph.
• Not using the correct formula to find the x-coordinate of the vertex.
• Not substituting the x-coordinate of the vertex into the function to find the y-coordinate of the vertex.
• Not using the correct formula to find the axis of symmetry.
• Not setting the function equal to zero to find the x-intercepts.

Q: How do I troubleshoot common mistakes when graphing quadratic functions?

A: To troubleshoot common mistakes when graphing quadratic functions, you need to:

• Review the properties of the graph, such as the vertex, axis of symmetry, and x-intercepts.
• Check the formula used to find the x-coordinate of the vertex.
• Substitute the x-coordinate of the vertex into the function to find the y-coordinate of the vertex.
• Use the correct formula to find the axis of symmetry.
• Set the function equal to zero to find the x-intercepts.

Conclusion

In conclusion, graphing quadratic functions is a fundamental concept in mathematics, and it has several real-world applications. To graph a quadratic function, you need to identify its vertex, axis of symmetry, and x-intercepts. You can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then substitute this value into the function to find the y-coordinate of the vertex. The graph of a quadratic function has several properties, including a vertex, axis of symmetry, x-intercepts, domain, and range. The graph has several real-world applications, including physics, engineering, and economics.