Identifying the Vertex of the Original Function

Oct 13, 2025 by magicalceylontours.com 0 views

Understanding the Original Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$ . In this case, the function is $f(x) = x^2 + 22x + 58$ . 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.

Identifying the Vertex of the Original Function

To find the vertex of the original function, we can use the formula $h = -\frac{b}{2a}$ and $k = f(h)$ . In this case, $a = 1$ and $b = 22$ . Plugging these values into the formula, we get $h = -\frac{22}{2(1)} = -11$ . To find $k$ , we plug $h = -11$ into the original function: $f(-11) = (-11)^2 + 22(-11) + 58 = 121 - 242 + 58 = -63$ . Therefore, the vertex of the original function is $(-11, -63)$ .

Translating the Function

The function is translated 4 units to the right and 16 units up. This means that the new vertex will be 4 units to the right and 16 units up from the original vertex. Therefore, the new vertex is $(-11 + 4, -63 + 16) = (-7, -47)$ .

Finding the Vertex Form of the New Function

Now that we have the new vertex, we can find the vertex form of the new function. The vertex form is given by $f(x) = a(x - h)^2 + k$ . In this case, $a = 1$ , $h = -7$ , and $k = -47$ . Plugging these values into the formula, we get $f(x) = (x - (-7))^2 + (-47) = (x + 7)^2 - 47$ .

Conclusion

The function $f(x) = x^2 + 22x + 58$ is translated 4 units to the right and 16 units up. The vertex form of the new function is $(x + 7)^2 - 47$ .

Answer

The correct answer is C. $(x + 7)^2 - 47$ .

Explanation

The other options are incorrect because they do not accurately reflect the translation of the function. Option A is incorrect because it does not account for the translation to the right. Option B is incorrect because it does not account for the translation up. Option D is not a valid option.

Example

To illustrate the concept, let's consider an example. Suppose we have a function $f(x) = x^2 + 6x + 8$ . If we translate this function 3 units to the right and 2 units up, the new function will be $f(x) = (x - 3)^2 + 2 + 8 = (x - 3)^2 + 10$ .

Tips and Tricks

When translating a function, make sure to account for both the horizontal and vertical translations.
Use the formula $h = -\frac{b}{2a}$ and $k = f(h)$ to find the vertex of the original function.
Use the formula $f(x) = a(x - h)^2 + k$ to find the vertex form of the new function.
Make sure to plug in the correct values for $a$ , $h$ , and $k$ into the formula.

Practice Problems

Translate the function $f(x) = x^2 + 4x + 9$ 2 units to the right and 1 unit up. Find the vertex form of the new function.
Translate the function $f(x) = x^2 - 2x + 1$ 1 unit to the left and 3 units down. Find the vertex form of the new function.

Conclusion

In conclusion, translating a function involves both horizontal and vertical translations. To find the vertex form of the new function, we need to find the new vertex and plug in the correct values into the formula. With practice and patience, you can master the art of function translation and vertex form.

Q: What is function translation?

A: Function translation is the process of changing the position of a function on the coordinate plane. This can involve shifting the function to the left or right (horizontal translation) or up or down (vertical translation).

Q: How do I translate a function?

A: To translate a function, you need to identify the type of translation (horizontal or vertical) and the amount of translation. For horizontal translations, you add or subtract a value from the x-coordinate of the function. For vertical translations, you add or subtract a value from the y-coordinate of the function.

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

A: To find the vertex of a quadratic function, you can use the formula $h = -\frac{b}{2a}$ and $k = f(h)$ . This will give you the x-coordinate and y-coordinate of the vertex, respectively.

Q: What is the difference between a horizontal translation and a vertical translation?

A: A horizontal translation involves shifting the function to the left or right, while a vertical translation involves shifting the function up or down.

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

A: To find the vertex form of a translated function, you need to find the new vertex and plug in the correct values into the formula $f(x) = a(x - h)^2 + k$ .

Q: Can I translate a function more than once?

A: Yes, you can translate a function more than once. Each translation will result in a new function with a new vertex.

Q: How do I determine the direction of a translation?

A: To determine the direction of a translation, you need to look at the sign of the translation value. A positive value indicates a translation to the right or up, while a negative value indicates a translation to the left or down.

Q: Can I use function translation to solve real-world problems?

A: Yes, function translation can be used to solve real-world problems. For example, you can use function translation to model the motion of an object or the growth of a population.

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

A: Some common mistakes to avoid when translating functions include:

Not accounting for both horizontal and vertical translations
Not using the correct formula for the vertex form
Not plugging in the correct values into the formula
Not checking the direction of the translation

Q: How can I practice function translation?

A: You can practice function translation by working through examples and exercises. You can also use online resources and tools to help you visualize and understand the concept of function translation.

Q: What are some real-world applications of function translation?

A: Some real-world applications of function translation include:

Modeling the motion of an object
Modeling the growth of a population
Modeling the spread of a disease
Modeling the behavior of a financial system

Q: Can I use function translation to solve optimization problems?

A: Yes, function translation can be used to solve optimization problems. For example, you can use function translation to find the maximum or minimum value of a function.

Q: What are some common types of function translations?

A: Some common types of function translations include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in calculus?

A: You can use function translation to solve problems in calculus by applying the concept of function translation to optimization problems, such as finding the maximum or minimum value of a function.

Q: What are some common mistakes to avoid when using function translation in calculus?

A: Some common mistakes to avoid when using function translation in calculus include:

Not accounting for both horizontal and vertical translations
Not using the correct formula for the vertex form
Not plugging in the correct values into the formula
Not checking the direction of the translation

Q: How can I use function translation to solve problems in physics?

A: You can use function translation to solve problems in physics by applying the concept of function translation to model the motion of an object or the growth of a population.

Q: What are some common types of function translations in physics?

A: Some common types of function translations in physics include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in engineering?

A: You can use function translation to solve problems in engineering by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in engineering?

A: Some common types of function translations in engineering include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in economics?

A: You can use function translation to solve problems in economics by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in economics?

A: Some common types of function translations in economics include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in computer science?

A: You can use function translation to solve problems in computer science by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in computer science?

A: Some common types of function translations in computer science include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in data analysis?

A: You can use function translation to solve problems in data analysis by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in data analysis?

A: Some common types of function translations in data analysis include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in machine learning?

A: You can use function translation to solve problems in machine learning by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in machine learning?

A: Some common types of function translations in machine learning include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in natural language processing?

A: You can use function translation to solve problems in natural language processing by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in natural language processing?

A: Some common types of function translations in natural language processing include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in computer vision?

A: You can use function translation to solve problems in computer vision by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in computer vision?

A: Some common types of function translations in computer vision include:

Horizontal translations (shifting to the left or right)
Vertical translations (shifting up or down)
Reflections (flipping over the x-axis or y-axis)
Rotations (rotating around the origin)

Q: How can I use function translation to solve problems in robotics?

A: You can use function translation to solve problems in robotics by applying the concept of function translation to model the behavior of a system or the growth of a population.

Q: What are some common types of function translations in robotics?

A: Some common types of function translations in robotics include:

Horizontal translations (shifting to the left or right)