Which Of The Following Is The Equation Of $d(x)$ In Terms Of $p(x)$? Explain Your Reasoning.A. $d(x)=p(x+3)+2$B. $ D ( X ) = P ( X − 3 ) + 2 D(x)=p(x-3)+2 D ( X ) = P ( X − 3 ) + 2 [/tex]C. $d(x)=p(x+3)-2$D. $d(x)=p(x-3)-2$

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Understanding the Relationship Between d(x) and p(x)

When dealing with functions, it's essential to understand how they relate to each other. In this case, we're given a function $p(x)$ and asked to find the equation of $d(x)$ in terms of $p(x)$. To do this, we need to analyze the given options and determine which one correctly represents the relationship between $d(x)$ and $p(x)$.

Analyzing the Options

Let's start by analyzing each option and understanding what it implies.

Option A: $d(x)=p(x+3)+2$

This option suggests that the function $d(x)$ is equal to $p(x+3)$ shifted 3 units to the left and then shifted 2 units upwards. This means that the graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.

Option B: $d(x)=p(x-3)+2$

This option suggests that the function $d(x)$ is equal to $p(x-3)$ shifted 3 units to the right and then shifted 2 units upwards. This means that the graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the right and 2 units upwards.

Option C: $d(x)=p(x+3)-2$

This option suggests that the function $d(x)$ is equal to $p(x+3)$ shifted 3 units to the left and then shifted 2 units downwards. This means that the graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units downwards.

Option D: $d(x)=p(x-3)-2$

This option suggests that the function $d(x)$ is equal to $p(x-3)$ shifted 3 units to the right and then shifted 2 units downwards. This means that the graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the right and 2 units downwards.

Determining the Correct Option

To determine the correct option, we need to understand the concept of function transformations. When a function is shifted horizontally, the input value changes, but the output value remains the same. When a function is shifted vertically, the output value changes, but the input value remains the same.

In this case, we're given a function $p(x)$ and asked to find the equation of $d(x)$ in terms of $p(x)$. The correct option should reflect a horizontal shift of 3 units and a vertical shift of 2 units.

Conclusion

Based on our analysis, we can conclude that the correct option is:

A. $d(x)=p(x+3)+2$

This option correctly reflects a horizontal shift of 3 units and a vertical shift of 2 units. The graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.

Understanding the Reasoning

To understand the reasoning behind this conclusion, let's consider the following:

  • When a function is shifted horizontally, the input value changes, but the output value remains the same. In this case, the input value changes from $x$ to $x+3$, which means that the function is shifted 3 units to the left.
  • When a function is shifted vertically, the output value changes, but the input value remains the same. In this case, the output value changes by adding 2, which means that the function is shifted 2 units upwards.

Therefore, the correct option is the one that reflects a horizontal shift of 3 units and a vertical shift of 2 units, which is option A: $d(x)=p(x+3)+2$.

Key Takeaways

  • When dealing with functions, it's essential to understand how they relate to each other.
  • To determine the correct option, we need to analyze the given options and understand what they imply.
  • The correct option should reflect a horizontal shift of 3 units and a vertical shift of 2 units.
  • The graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.

By understanding these key takeaways, we can confidently conclude that the correct option is A: $d(x)=p(x+3)+2$.
Q&A: Understanding the Relationship Between d(x) and p(x)

In our previous article, we discussed the relationship between the functions $d(x)$ and $p(x)$. We analyzed the given options and determined that the correct option is A: $d(x)=p(x+3)+2$. In this article, we'll answer some frequently asked questions to further clarify the relationship between $d(x)$ and $p(x)$.

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

A: A horizontal shift occurs when the input value changes, but the output value remains the same. In the case of $d(x)=p(x+3)+2$, the input value changes from $x$ to $x+3$, which means that the function is shifted 3 units to the left. A vertical shift occurs when the output value changes, but the input value remains the same. In this case, the output value changes by adding 2, which means that the function is shifted 2 units upwards.

Q: Why is the correct option A: $d(x)=p(x+3)+2$?

A: The correct option is A: $d(x)=p(x+3)+2$ because it reflects a horizontal shift of 3 units and a vertical shift of 2 units. The graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.

Q: What is the significance of the +3 in the correct option?

A: The +3 in the correct option represents a horizontal shift of 3 units to the left. This means that the input value changes from $x$ to $x+3$, which shifts the graph of $p(x)$ 3 units to the left.

Q: What is the significance of the +2 in the correct option?

A: The +2 in the correct option represents a vertical shift of 2 units upwards. This means that the output value changes by adding 2, which shifts the graph of $p(x)$ 2 units upwards.

Q: How can I visualize the relationship between $d(x)$ and $p(x)$?

A: You can visualize the relationship between $d(x)$ and $p(x)$ by graphing both functions on the same coordinate plane. The graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.

Q: What are some common mistakes to avoid when working with function transformations?

A: Some common mistakes to avoid when working with function transformations include:

  • Confusing horizontal and vertical shifts
  • Failing to account for the direction of the shift (left or right, up or down)
  • Not considering the effect of the shift on the graph of the function

Q: How can I apply this knowledge to real-world problems?

A: You can apply this knowledge to real-world problems by considering the following:

  • When analyzing data, consider how the data has been transformed (e.g. shifted, scaled, etc.)
  • When modeling real-world phenomena, consider how the model has been transformed (e.g. shifted, scaled, etc.)
  • When communicating results, consider how the results have been transformed (e.g. shifted, scaled, etc.)

By understanding the relationship between $d(x)$ and $p(x)$ and applying this knowledge to real-world problems, you can develop a deeper understanding of function transformations and their applications.

Key Takeaways

  • A horizontal shift occurs when the input value changes, but the output value remains the same.
  • A vertical shift occurs when the output value changes, but the input value remains the same.
  • The correct option is A: $d(x)=p(x+3)+2$ because it reflects a horizontal shift of 3 units and a vertical shift of 2 units.
  • The graph of $d(x)$ will be the same as the graph of $p(x)$, but shifted 3 units to the left and 2 units upwards.
  • You can visualize the relationship between $d(x)$ and $p(x)$ by graphing both functions on the same coordinate plane.

By understanding these key takeaways, you can confidently apply this knowledge to real-world problems and develop a deeper understanding of function transformations.