Find The Coordinates Of The Vertex For The Parabola Defined By The Given Quadratic Function. F ( X ) = ( X + 3 ) 2 + 5 F(x)=(x+3)^2+5 F ( X ) = ( X + 3 ) 2 + 5 A. (5, -9)B. (-3, 5)C. (5, -3)D. (-5, 3)

Mar 13, 2025 by ADMIN 201 views

**Finding the Coordinates of the Vertex for a Parabola Defined by a Quadratic Function**

Understanding the Problem

When dealing with quadratic functions, one of the key concepts is the vertex of the parabola. The vertex is the highest or lowest point on the parabola, and it plays a crucial role in understanding the behavior of the function. In this article, we will focus on finding the coordinates of the vertex for a parabola defined by the given quadratic function, $f(x)=(x+3)^2+5$ .

The General Form of a Quadratic Function

A quadratic function can be written in the general form $f(x)=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants. However, in this case, we are given a quadratic function in the form $f(x)=(x+3)^2+5$ . This form is known as the vertex form of a quadratic function, where the vertex is given by the point $(-3, 5)$ .

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, we have $f(x)=(x+3)^2+5$ , which can be rewritten as $f(x)=(x-(-3))^2+5$ . Therefore, the vertex of the parabola is given by the point $(-3, 5)$ .

Finding the Coordinates of the Vertex

To find the coordinates of the vertex, we need to identify the values of $h$ and $k$ in the vertex form of the quadratic function. In this case, we have $h=-3$ and $k=5$ . Therefore, the coordinates of the vertex are given by the point $(-3, 5)$ .

Comparing the Options

Now that we have found the coordinates of the vertex, we can compare them with the options given in the problem. The options are:

A. (5, -9) B. (-3, 5) C. (5, -3) D. (-5, 3)

Comparing the coordinates of the vertex with the options, we can see that the correct answer is:

B. (-3, 5)

Conclusion

In this article, we have discussed how to find the coordinates of the vertex for a parabola defined by a quadratic function. We have used the vertex form of a quadratic function to identify the vertex of the parabola and have compared the coordinates of the vertex with the options given in the problem. The correct answer is B. (-3, 5).

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.
To find the coordinates of the vertex, we need to identify the values of $h$ and $k$ in the vertex form of the quadratic function.
The vertex of the parabola is given by the point $(-3, 5)$ .

Final Answer

The final answer is B. (-3, 5).

Understanding the Problem

The General Form of a Quadratic Function

The Vertex Form of a Quadratic Function

Finding the Coordinates of the Vertex

Comparing the Options

Now that we have found the coordinates of the vertex, we can compare them with the options given in the problem. The options are:

A. (5, -9) B. (-3, 5) C. (5, -3) D. (-5, 3)

Comparing the coordinates of the vertex with the options, we can see that the correct answer is:

B. (-3, 5)

Conclusion

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.
To find the coordinates of the vertex, we need to identify the values of $h$ and $k$ in the vertex form of the quadratic function.
The vertex of the parabola is given by the point $(-3, 5)$ .

Final Answer

The final answer is B. (-3, 5).

Quadratic Function Vertex: Frequently Asked Questions

Q1: What is the vertex form of a quadratic function?

A1: 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.

Q2: How do I find the coordinates of the vertex?

A2: To find the coordinates of the vertex, we need to identify the values of $h$ and $k$ in the vertex form of the quadratic function.

Q3: What is the significance of the vertex in a quadratic function?

A3: The vertex is the highest or lowest point on the parabola, and it plays a crucial role in understanding the behavior of the function.

Q4: Can I use the general form of a quadratic function to find the vertex?

A4: Yes, you can use the general form of a quadratic function to find the vertex. However, it may be more complicated than using the vertex form.

Q5: How do I compare the coordinates of the vertex with the options given in the problem?

A5: To compare the coordinates of the vertex with the options, you need to identify the values of $h$ and $k$ in the vertex form of the quadratic function and compare them with the options.

Q6: What is the final answer to the problem?

A6: The final answer is B. (-3, 5).

Q7: Can I use a calculator to find the vertex of a quadratic function?

A7: Yes, you can use a calculator to find the vertex of a quadratic function. However, it is always a good idea to understand the concept behind the calculation.

Q8: How do I graph a quadratic function with a given vertex?

A8: To graph a quadratic function with a given vertex, you need to use the vertex form of the quadratic function and plot the point $(h, k)$ on the graph.

Q9: Can I use the vertex form of a quadratic function to find the equation of the parabola?

A9: Yes, you can use the vertex form of a quadratic function to find the equation of the parabola. Simply plug in the values of $h$ and $k$ into the vertex form of the quadratic function.

Q10: How do I use the vertex form of a quadratic function to find the x-intercepts of the parabola?

A10: To find the x-intercepts of the parabola, you need to set the function equal to zero and solve for $x$ .