Which Describes The Graph Of $y=-(x+6)^2+6$?A. Maximum At $(6,6$\] B. Maximum At $(-6,6$\] C. Minimum At $(6,6$\] D. Minimum At $(-6,6$\]

Introduction

In mathematics, 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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on the graph of the quadratic function $y=-(x+6)^2+6$ and determine its characteristics.

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. In this case, the given function is $y=-(x+6)^2+6$, which can be rewritten as $y=-(x-(-6))^2+6$. Therefore, the vertex of the parabola is $(-6,6)$.

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 $x=-6$.

The Maximum or Minimum Value

Since the coefficient of the squared term is negative, the parabola opens downward, and the vertex represents the maximum value of the function. Therefore, the maximum value of the function is $6$, which occurs at the vertex $(-6,6)$.

Comparing the Options

Now, let's compare the options given in the problem:

• A. Maximum at $(6,6)$: This option is incorrect because the vertex is at $(-6,6)$, not $(6,6)$.
• B. Maximum at $(-6,6)$: This option is correct because the vertex is at $(-6,6)$, and the parabola opens downward.
• C. Minimum at $(6,6)$: This option is incorrect because the vertex is at $(-6,6)$, not $(6,6)$, and the parabola opens downward, indicating a maximum value.
• D. Minimum at $(-6,6)$: This option is incorrect because the vertex is at $(-6,6)$, and the parabola opens downward, indicating a maximum value.

Conclusion

In conclusion, the graph of the quadratic function $y=-(x+6)^2+6$ has a maximum value of $6$ at the vertex $(-6,6)$. Therefore, the correct answer is option B.

Key Takeaways

• 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.
• The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola.
• The maximum or minimum value of a quadratic function occurs at the vertex of the parabola.
• If the coefficient of the squared term is negative, the parabola opens downward, and the vertex represents the maximum value of the function.

Further Reading

For more information on quadratic functions and their graphs, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• Larson, R., & Hostetler, R. P. (2015). College Algebra: A Concise Approach. Cengage Learning.
• Sullivan, M. (2016). College Algebra. Pearson Education.

Introduction

In our previous article, we discussed the graph of the quadratic function $y=-(x+6)^2+6$ and determined its characteristics. In this article, we will provide a Q&A guide to help you better understand quadratic function graphs.

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

A: 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 parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola.

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

A: To determine the maximum or minimum value of a quadratic function, you need to look at the coefficient of the squared term. If the coefficient is positive, the parabola opens upward, and the vertex represents the minimum value of the function. If the coefficient is negative, the parabola opens downward, and the vertex represents the maximum value of the function.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola represents the maximum or minimum value of the function, depending on whether the parabola opens upward or downward.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

1. Determine the vertex of the parabola.
2. Draw a vertical line through the vertex, which represents the axis of symmetry.
3. Choose a point on one side of the axis of symmetry and find the corresponding point on the other side of the axis of symmetry.
4. Plot the points and draw a smooth curve through them.

Q: What are some common types of quadratic functions?

A: Some common types of quadratic functions include:

• Linear functions: These functions have a constant slope and a single variable.
• Quadratic functions: These functions have a squared variable and a single variable.
• Cubic functions: These functions have a cubed variable and a single variable.

Q: How do I determine the domain and range of a quadratic function?

A: To determine the domain and range of a quadratic function, you need to look at the vertex and the axis of symmetry. The domain of a quadratic function is all real numbers, and the range is determined by the vertex and the axis of symmetry.

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

A: Quadratic functions have many real-world applications, including:

• Projectile motion: Quadratic functions can be used to model the trajectory of a projectile.
• Optimization problems: Quadratic functions can be used to solve optimization problems, such as finding the maximum or minimum value of a function.
• Electrical circuits: Quadratic functions can be used to model the behavior of electrical circuits.

Conclusion

In conclusion, quadratic function graphs are an important topic in mathematics, and understanding them can help you solve a wide range of problems. By following the steps outlined in this article, you can better understand quadratic function graphs and apply them to real-world problems.

Key Takeaways

• 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.
• The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola.
• The maximum or minimum value of a quadratic function occurs at the vertex of the parabola.
• Quadratic functions have many real-world applications, including projectile motion, optimization problems, and electrical circuits.

Further Reading

For more information on quadratic function graphs, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• Larson, R., & Hostetler, R. P. (2015). College Algebra: A Concise Approach. Cengage Learning.
• Sullivan, M. (2016). College Algebra. Pearson Education.

Note: The references provided are for general information purposes only and are not specific to the topic of this article.