(c)Given:$\[ \begin{align*} f(x) &= 2^x \\ c(x) &= 4f(x) \end{align*} \\]Complete The Table With Corresponding Points:$\[ \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Reference \\ Points On $f(x)$ \end{tabular} & \begin{tabular}{c}

Given the functions $f(x) = 2^x$ and $c(x) = 4f(x)$, we are tasked with completing a table with corresponding points. To do this, we need to understand the relationship between the two functions and how they interact with each other.

Understanding the Relationship Between the Functions

The function $f(x) = 2^x$ is an exponential function that takes a base of 2 and raises it to the power of $x$. This function is increasing and has a horizontal asymptote at $y = 0$. On the other hand, the function $c(x) = 4f(x)$ is a transformation of the function $f(x)$ that involves a vertical stretch by a factor of 4.

Calculating the Corresponding Points

To complete the table with corresponding points, we need to find the values of $x$ and $y$ for each point. We can do this by substituting the values of $x$ into the function $f(x) = 2^x$ and then multiplying the result by 4 to get the corresponding value of $y$ for the function $c(x)$.

Calculating the Corresponding Points for $x = -2$

For $x = -2$, we have:

$f(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4}$

$c(-2) = 4f(-2) = 4 \times \frac{1}{4} = 1$

So, the corresponding point for $x = -2$ is $\left(-2, 1\right)$.

Calculating the Corresponding Points for $x = 0$

For $x = 0$, we have:

$f(0) = 2^0 = 1$

$c(0) = 4f(0) = 4 \times 1 = 4$

So, the corresponding point for $x = 0$ is $\left(0, 4\right)$.

Calculating the Corresponding Points for $x = 2$

For $x = 2$, we have:

$f(2) = 2^2 = 4$

$c(2) = 4f(2) = 4 \times 4 = 16$

So, the corresponding point for $x = 2$ is $\left(2, 16\right)$.

Calculating the Corresponding Points for $x = 3$

For $x = 3$, we have:

$f(3) = 2^3 = 8$

$c(3) = 4f(3) = 4 \times 8 = 32$

So, the corresponding point for $x = 3$ is $\left(3, 32\right)$.

Calculating the Corresponding Points for $x = 4$

For $x = 4$, we have:

$f(4) = 2^4 = 16$

$c(4) = 4f(4) = 4 \times 16 = 64$

So, the corresponding point for $x = 4$ is $\left(4, 64\right)$.

Completing the Table

Based on the calculations above, we can complete the table with corresponding points as follows:

Reference Points on $f(x)$	Discussion Category: Mathematics
$\left(-2, 1\right)$
$\left(0, 4\right)$
$\left(2, 16\right)$
$\left(3, 32\right)$
$\left(4, 64\right)$

Conclusion

In this article, we have completed a table with corresponding points for the functions $f(x) = 2^x$ and $c(x) = 4f(x)$. We have calculated the values of $x$ and $y$ for each point by substituting the values of $x$ into the function $f(x)$ and then multiplying the result by 4 to get the corresponding value of $y$ for the function $c(x)$. The completed table provides a visual representation of the relationship between the two functions and can be used as a reference for further mathematical exploration.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-26.
• [2] "Transformations of Functions". Math Is Fun. Retrieved 2023-02-26.

Keywords

• Exponential functions
• Transformations of functions
• Corresponding points
• Mathematical exploration
  Q&A: Completing the Table with Corresponding Points =====================================================

In the previous article, we completed a table with corresponding points for the functions $f(x) = 2^x$ and $c(x) = 4f(x)$. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the relationship between the functions $f(x)$ and $c(x)$?

A: The function $c(x)$ is a transformation of the function $f(x)$ that involves a vertical stretch by a factor of 4. This means that for every value of $x$, the corresponding value of $y$ for the function $c(x)$ is 4 times the value of $y$ for the function $f(x)$.

Q: How do I calculate the corresponding points for the function $c(x)$?

A: To calculate the corresponding points for the function $c(x)$, you need to substitute the values of $x$ into the function $f(x) = 2^x$ and then multiply the result by 4 to get the corresponding value of $y$ for the function $c(x)$.

Q: What is the significance of the table with corresponding points?

A: The table with corresponding points provides a visual representation of the relationship between the two functions and can be used as a reference for further mathematical exploration.

Q: Can I use the table with corresponding points to solve problems involving the functions $f(x)$ and $c(x)$?

A: Yes, you can use the table with corresponding points to solve problems involving the functions $f(x)$ and $c(x)$. For example, you can use the table to find the value of $y$ for a given value of $x$ for either function.

Q: How do I extend the table with corresponding points to include more values of $x$?

A: To extend the table with corresponding points to include more values of $x$, you need to substitute the new values of $x$ into the function $f(x) = 2^x$ and then multiply the result by 4 to get the corresponding value of $y$ for the function $c(x)$.

Q: Can I use the table with corresponding points to explore other mathematical concepts?

A: Yes, you can use the table with corresponding points to explore other mathematical concepts, such as transformations of functions, exponential functions, and mathematical modeling.

Q: How do I use the table with corresponding points to solve real-world problems?

A: You can use the table with corresponding points to solve real-world problems by applying the mathematical concepts and relationships represented in the table to real-world scenarios.

Conclusion

In this article, we have answered some frequently asked questions related to completing the table with corresponding points for the functions $f(x) = 2^x$ and $c(x) = 4f(x)$. We hope that this article has provided you with a better understanding of the topic and has helped you to develop your problem-solving skills.

References

• [1] "Exponential Functions". Math Open Reference. Retrieved 2023-02-26.
• [2] "Transformations of Functions". Math Is Fun. Retrieved 2023-02-26.

Keywords

• Exponential functions
• Transformations of functions
• Corresponding points
• Mathematical exploration
• Problem-solving skills