Find $g(x)$, Where $g(x)$ Is The Translation 7 Units Up Of $f(x) = X^2$. Write Your Answer In The Form \$a(x - H)^2 + K$[/tex\], Where $a$, $h$, And \$k$[/tex\] Are

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Introduction

In mathematics, a translation of a function is a transformation that moves the graph of the function to a new position. In this article, we will explore how to find the translation of a given function, specifically the translation 7 units up of the function $f(x) = x^2$. We will use the vertex form of a quadratic function, which is given by $a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants.

Understanding the Problem

The problem asks us to find the translation 7 units up of the function $f(x) = x^2$. This means that we need to shift the graph of $f(x)$ 7 units up to obtain the new function $g(x)$. To do this, we will use the vertex form of a quadratic function and apply the translation to the vertex of the parabola.

Translation of a Function

A translation of a function is a transformation that moves the graph of the function to a new position. There are two types of translations: horizontal and vertical. A horizontal translation moves the graph of the function to the left or right, while a vertical translation moves the graph of the function up or down.

In this case, we are dealing with a vertical translation, which moves the graph of the function up or down. The translation 7 units up means that we need to add 7 to the function $f(x)$ to obtain the new function $g(x)$.

Finding $g(x)$

To find $g(x)$, we need to add 7 to the function $f(x) = x^2$. This can be done by adding 7 to the expression for $f(x)$:

g(x)=f(x)+7=x2+7g(x) = f(x) + 7 = x^2 + 7

However, we are asked to write the answer in the form $a(x - h)^2 + k$. To do this, we need to complete the square for the expression $x^2 + 7$.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form $a(x - h)^2 + k$. To complete the square for the expression $x^2 + 7$, we need to add and subtract the square of half the coefficient of $x^2$.

The coefficient of $x^2$ is 1, so half of this coefficient is $\frac{1}{2}$. The square of $\frac{1}{2}$ is $\frac{1}{4}$.

Adding and subtracting $\frac{1}{4}$ to the expression $x^2 + 7$ gives:

x2+7=(x2+14)+7−14x^2 + 7 = (x^2 + \frac{1}{4}) + 7 - \frac{1}{4}

Simplifying this expression gives:

x2+7=(x2+14)+274x^2 + 7 = (x^2 + \frac{1}{4}) + \frac{27}{4}

Now, we can factor the perfect square trinomial:

x2+14=(x+12)2x^2 + \frac{1}{4} = (x + \frac{1}{2})^2

Substituting this back into the expression gives:

g(x)=(x+12)2+274g(x) = (x + \frac{1}{2})^2 + \frac{27}{4}

Conclusion

In this article, we have found the translation 7 units up of the function $f(x) = x^2$. We have used the vertex form of a quadratic function and applied the translation to the vertex of the parabola. We have also completed the square to rewrite the expression in the form $a(x - h)^2 + k$.

The final answer is:

g(x)=(x+12)2+274g(x) = (x + \frac{1}{2})^2 + \frac{27}{4}

Key Takeaways

  • A translation of a function is a transformation that moves the graph of the function to a new position.
  • There are two types of translations: horizontal and vertical.
  • A vertical translation moves the graph of the function up or down.
  • Completing the square is a technique used to rewrite a quadratic expression in the form $a(x - h)^2 + k$.
  • The vertex form of a quadratic function is given by $a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants.

Further Reading

Introduction

In our previous article, we explored how to find the translation of a given function, specifically the translation 7 units up of the function $f(x) = x^2$. We used the vertex form of a quadratic function and applied the translation to the vertex of the parabola. In this article, we will answer some common questions related to translations of functions.

Q: What is a translation of a function?

A: A translation of a function is a transformation that moves the graph of the function to a new position. There are two types of translations: horizontal and vertical. A horizontal translation moves the graph of the function to the left or right, while a vertical translation moves the graph of the function up or down.

Q: How do I find the translation of a function?

A: To find the translation of a function, you need to apply the translation to the vertex of the parabola. For a vertical translation, you add or subtract the translation value from the function. For a horizontal translation, you add or subtract the translation value from the x-coordinate of the vertex.

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

A: The vertex form of a quadratic function is given by $a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants. The vertex of the parabola is given by the point $(h, k)$.

Q: How do I complete the square to rewrite a quadratic expression in the form $a(x - h)^2 + k$?

A: To complete the square, you need to add and subtract the square of half the coefficient of $x^2$. The coefficient of $x^2$ is 1, so half of this coefficient is $\frac{1}{2}$. The square of $\frac{1}{2}$ is $\frac{1}{4}$. Adding and subtracting $\frac{1}{4}$ to the expression $x^2 + 7$ gives:

x2+7=(x2+14)+7−14x^2 + 7 = (x^2 + \frac{1}{4}) + 7 - \frac{1}{4}

Simplifying this expression gives:

x2+7=(x2+14)+274x^2 + 7 = (x^2 + \frac{1}{4}) + \frac{27}{4}

Now, you can factor the perfect square trinomial:

x2+14=(x+12)2x^2 + \frac{1}{4} = (x + \frac{1}{2})^2

Substituting this back into the expression gives:

g(x)=(x+12)2+274g(x) = (x + \frac{1}{2})^2 + \frac{27}{4}

Q: What are some common types of translations?

A: Some common types of translations include:

  • Vertical translation: moving the graph of the function up or down
  • Horizontal translation: moving the graph of the function to the left or right
  • Reflection: flipping the graph of the function over a line or axis
  • Rotation: rotating the graph of the function around a point or axis

Q: How do I apply a translation to a function?

A: To apply a translation to a function, you need to add or subtract the translation value from the function. For a vertical translation, you add or subtract the translation value from the function. For a horizontal translation, you add or subtract the translation value from the x-coordinate of the vertex.

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

A: Translations of functions have many real-world applications, including:

  • Physics: translations of functions are used to model the motion of objects
  • Engineering: translations of functions are used to design and optimize systems
  • Computer Science: translations of functions are used in algorithms and data structures

Conclusion

In this article, we have answered some common questions related to translations of functions. We have discussed the vertex form of a quadratic function, completing the square, and applying translations to functions. We have also explored some real-world applications of translations of functions.

Key Takeaways

  • A translation of a function is a transformation that moves the graph of the function to a new position.
  • There are two types of translations: horizontal and vertical.
  • Completing the square is a technique used to rewrite a quadratic expression in the form $a(x - h)^2 + k$.
  • The vertex form of a quadratic function is given by $a(x - h)^2 + k$, where $a$, $h$, and $k$ are constants.
  • Translations of functions have many real-world applications, including physics, engineering, and computer science.

Further Reading