Which Describes The Graph Of $y=(x+6)^2+1$?A. Vertex At \[$(-6, -1)\$\]B. Vertex At \[$(-6, 1)\$\]C. Vertex At \[$(6, 1)\$\]D. Vertex At \[$(5, -1)\$\]

Feb 28, 2025 by ADMIN 152 views

**Understanding the Graph of a Quadratic Function**

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 $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this article, we will focus on the graph of the quadratic function $y = (x + 6)^2 + 1$ . We will analyze the characteristics of this graph and determine which of the given options describes it accurately.

The Standard Form of a Quadratic Function

The standard form of a quadratic function is $y = ax^2 + bx + c$ . However, the given function $y = (x + 6)^2 + 1$ is in the form of a vertex form, which is $y = a(x - h)^2 + k$ . In this form, the vertex of the parabola is at the point $(h, k)$ . To convert the given function to the standard form, we can expand the squared term:

y = (x + 6)^2 + 1

y = x^2 + 12x + 36 + 1

y = x^2 + 12x + 37

Identifying the Vertex

Now that we have the standard form of the quadratic function, we can identify the vertex. The vertex is the point at which the parabola changes direction. In the standard form, the vertex is at the point $(h, k)$ . Comparing the given function with the standard form, we can see that $h = -6$ and $k = 1$ . Therefore, the vertex of the parabola is at the point $(-6, 1)$ .

Analyzing the Options

Now that we have identified the vertex, we can analyze the options given in the problem. The options are:

A. Vertex at $(-6, -1)$ B. Vertex at $(-6, 1)$ C. Vertex at $(6, 1)$ D. Vertex at $(5, -1)$

Based on our analysis, we can see that option B is the correct answer. The vertex of the parabola is indeed at the point $(-6, 1)$ .

Conclusion

In conclusion, the graph of the quadratic function $y = (x + 6)^2 + 1$ has a vertex at the point $(-6, 1)$ . This is because the function is in the vertex form, and the vertex is at the point $(h, k)$ . We can convert the function to the standard form to identify the vertex, and then analyze the options to determine which one is correct. The correct answer is option B.

Key Takeaways

The vertex form of a quadratic function is $y = a(x - h)^2 + k$ .
The vertex of a parabola is at the point $(h, k)$ .
To identify the vertex, we can convert the function to the standard form.
The correct answer is option B.

References

[1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-functions
[2] Math Is Fun. Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
[3] Wolfram MathWorld. Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
Quadratic Function Graphs: A Q&A Guide =====================================

Introduction

In our previous article, we discussed the graph of the quadratic function $y = (x + 6)^2 + 1$ . We identified the vertex of the parabola and analyzed the options to determine which one was correct. In this article, we will provide a Q&A guide to help you better understand quadratic function graphs.

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 is two. The general form of a quadratic function is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

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

A: The vertex form of a quadratic function is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Q: How do I identify the vertex of a parabola?

A: To identify the vertex of a parabola, you can convert the function to the standard form $y = ax^2 + bx + c$ . The vertex is at the point $(h, k)$ , where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$ .

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. It is given by the equation $x = h$ , where $(h, k)$ is the vertex of the parabola.

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

A: To determine the direction of the parabola, you can look at the coefficient of the squared term. If $a > 0$ , the parabola opens upward. If $a < 0$ , the parabola opens downward.

Q: What is the x-intercept of a parabola?

A: The x-intercept of a parabola is the point where the parabola intersects the x-axis. It is given by the equation $y = 0$ .

Q: How do I find the x-intercept of a parabola?

A: To find the x-intercept of a parabola, you can set $y = 0$ and solve for $x$ . This will give you the x-coordinate of the x-intercept.

Q: What is the y-intercept of a parabola?

A: The y-intercept of a parabola is the point where the parabola intersects the y-axis. It is given by the equation $x = 0$ .

Q: How do I find the y-intercept of a parabola?

A: To find the y-intercept of a parabola, you can set $x = 0$ and solve for $y$ . This will give you the y-coordinate of the y-intercept.

Q: How do I graph a quadratic function?

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

Identify the vertex of the parabola.
Draw a vertical line through the vertex to represent the axis of symmetry.
Determine the direction of the parabola.
Plot the x-intercepts and y-intercepts.
Draw the parabola through the points.

Conclusion

In conclusion, quadratic function graphs are an important topic in mathematics. By understanding the properties of quadratic functions, you can graph them accurately and solve problems involving quadratic equations. We hope this Q&A guide has been helpful in your understanding of quadratic function graphs.

Key Takeaways

A quadratic function is a polynomial function of degree two.
The vertex form of a quadratic function is $y = a(x - h)^2 + k$ .
The axis of symmetry of a parabola is a vertical line that passes through the vertex.
The direction of the parabola is determined by the coefficient of the squared term.
The x-intercept of a parabola is the point where the parabola intersects the x-axis.
The y-intercept of a parabola is the point where the parabola intersects the y-axis.

References

[1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-equations/x2f-quadratic-functions
[2] Math Is Fun. Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
[3] Wolfram MathWorld. Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html

Which Describes The Graph Of $y=(x+6)^2+1$?A. Vertex At \[$(-6, -1)\$\]B. Vertex At \[$(-6, 1)\$\]C. Vertex At \[$(6, 1)\$\]D. Vertex At \[$(5, -1)\$\]

Introduction

The Standard Form of a Quadratic Function

Identifying the Vertex

Analyzing the Options

Conclusion

Key Takeaways

Further Reading

References

Introduction

Q: What is a quadratic function?

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

Q: How do I identify the vertex of a parabola?

Q: What is the axis of symmetry of a parabola?

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

Q: What is the x-intercept of a parabola?

Q: How do I find the x-intercept of a parabola?

Q: What is the y-intercept of a parabola?

Q: How do I find the y-intercept of a parabola?

Q: How do I graph a quadratic function?

Conclusion

Key Takeaways

Further Reading

References