Which Of These Functions Have A Graph That Is A Translation Of The Graph Of F ( X ) = X F(x)=x F ( X ) = X ? Select All That Apply.A. G ( X ) = 2 X G(x)=2x G ( X ) = 2 X B. H ( X ) = X + 1 H(x)=x+1 H ( X ) = X + 1 C. J ( X ) = X − 2 J(x)=x-2 J ( X ) = X − 2 D. K ( X ) = 1 2 X K(x)=\frac{1}{2}x K ( X ) = 2 1 ​ X

In mathematics, a translation is a transformation that moves a graph a certain distance in the horizontal or vertical direction. When we talk about the graph of $f(x)=x$, we are referring to a linear function that passes through the origin and has a slope of 1. In this article, we will explore which functions have a graph that is a translation of the graph of $f(x)=x$.

Understanding Translations

A translation is a type of transformation that moves a graph from one position to another without changing its shape or size. There are two types of translations: horizontal and vertical. A horizontal translation moves the graph to the left or right, while a vertical translation moves the graph up or down.

Horizontal Translations

A horizontal translation of the graph of $f(x)=x$ would result in a new function of the form $g(x)=x+c$, where $c$ is a constant. This means that the graph of $g(x)$ would be shifted $c$ units to the left or right of the graph of $f(x)$.

Vertical Translations

A vertical translation of the graph of $f(x)=x$ would result in a new function of the form $h(x)=f(x)+c$, where $c$ is a constant. This means that the graph of $h(x)$ would be shifted $c$ units up or down of the graph of $f(x)$.

Analyzing the Options

Now that we understand what a translation is, let's analyze the options given:

A. $g(x)=2x$

The graph of $g(x)=2x$ is a vertical stretch of the graph of $f(x)=x$. This is because the coefficient of $x$ is 2, which means that the graph is stretched vertically by a factor of 2. Therefore, the graph of $g(x)$ is not a translation of the graph of $f(x)$.

B. $h(x)=x+1$

The graph of $h(x)=x+1$ is a horizontal translation of the graph of $f(x)=x$. This is because the constant term is 1, which means that the graph is shifted 1 unit to the right of the graph of $f(x)$. Therefore, the graph of $h(x)$ is a translation of the graph of $f(x)$.

C. $j(x)=x-2$

The graph of $j(x)=x-2$ is a horizontal translation of the graph of $f(x)=x$. This is because the constant term is -2, which means that the graph is shifted 2 units to the left of the graph of $f(x)$. Therefore, the graph of $j(x)$ is a translation of the graph of $f(x)$.

D. $k(x)=\frac{1}{2}x$

The graph of $k(x)=\frac{1}{2}x$ is a vertical compression of the graph of $f(x)=x$. This is because the coefficient of $x$ is $\frac{1}{2}$, which means that the graph is compressed vertically by a factor of $\frac{1}{2}$. Therefore, the graph of $k(x)$ is not a translation of the graph of $f(x)$.

Conclusion

In conclusion, the functions that have a graph that is a translation of the graph of $f(x)=x$ are:

• $h(x)=x+1$
• $j(x)=x-2$

These functions are horizontal translations of the graph of $f(x)=x$. The other options, $g(x)=2x$ and $k(x)=\frac{1}{2}x$, are not translations of the graph of $f(x)$, but rather vertical stretches and compressions, respectively.

Key Takeaways

• A translation is a transformation that moves a graph a certain distance in the horizontal or vertical direction.
• A horizontal translation moves the graph to the left or right, while a vertical translation moves the graph up or down.
• The graph of $f(x)=x$ is a linear function that passes through the origin and has a slope of 1.
• The functions $h(x)=x+1$ and $j(x)=x-2$ are horizontal translations of the graph of $f(x)=x$.
• The functions $g(x)=2x$ and $k(x)=\frac{1}{2}x$ are not translations of the graph of $f(x)$, but rather vertical stretches and compressions, respectively.
  Q&A: Translations of the Graph of $f(x)=x$ =====================================================

In the previous article, we discussed the concept of translations and how they apply to the graph of $f(x)=x$. In this article, we will answer some frequently asked questions about translations of the graph of $f(x)=x$.

Q: What is a translation?

A translation is a transformation that moves a graph a certain distance in the horizontal or vertical direction. It does not change the shape or size of the graph, but rather its position.

Q: What are the two types of translations?

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

Q: How do I determine if a function is a translation of the graph of $f(x)=x$?

To determine if a function is a translation of the graph of $f(x)=x$, you need to look at the equation of the function. If the equation is of the form $g(x)=x+c$ or $h(x)=x+c$, where $c$ is a constant, then the graph of $g(x)$ or $h(x)$ is a horizontal translation of the graph of $f(x)=x$. If the equation is of the form $g(x)=f(x)+c$, then the graph of $g(x)$ is a vertical translation of the graph of $f(x)=x$.

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

A horizontal translation moves the graph to the left or right, while a vertical translation moves the graph up or down. For example, if we have the function $g(x)=x+1$, the graph of $g(x)$ is a horizontal translation of the graph of $f(x)=x$ because the constant term is 1. If we have the function $h(x)=x+2$, the graph of $h(x)$ is a vertical translation of the graph of $f(x)=x$ because the constant term is 2.

Q: Can a function be both a horizontal and vertical translation of the graph of $f(x)=x$?

No, a function cannot be both a horizontal and vertical translation of the graph of $f(x)=x$. A translation is a single transformation that moves the graph in one direction, either horizontally or vertically.

Q: How do I graph a translation of the graph of $f(x)=x$?

To graph a translation of the graph of $f(x)=x$, you need to follow these steps:

1. Graph the original function $f(x)=x$.
2. Determine the type of translation (horizontal or vertical).
3. Move the graph of $f(x)$ the required distance in the horizontal or vertical direction.
4. Label the new graph with the equation of the translated function.

Q: What are some examples of translations of the graph of $f(x)=x$?

Some examples of translations of the graph of $f(x)=x$ include:

• $g(x)=x+1$ (horizontal translation)
• $h(x)=x-2$ (horizontal translation)
• $j(x)=x+3$ (horizontal translation)
• $k(x)=x+2$ (vertical translation)
• $l(x)=x-1$ (vertical translation)

Conclusion

In conclusion, translations of the graph of $f(x)=x$ are an important concept in mathematics. By understanding how to determine if a function is a translation of the graph of $f(x)=x$, you can graph and analyze these functions with ease. Remember to follow the steps outlined above to graph a translation of the graph of $f(x)=x$.

Key Takeaways

• A translation is a transformation that moves a graph a certain distance in the horizontal or vertical direction.
• There are two types of translations: horizontal and vertical.
• To determine if a function is a translation of the graph of $f(x)=x$, look at the equation of the function.
• A horizontal translation moves the graph to the left or right, while a vertical translation moves the graph up or down.
• A function cannot be both a horizontal and vertical translation of the graph of $f(x)=x$.