Here are the solutions for the specified questions from the image:

Question 3

$f(x) = 3 \cdot 2^x$

• Power function check: A power function has the form $f(x) = kx^a$. This equation does not follow that form because the exponent $x$ is applied to a constant (2) rather than the variable $x$. Thus, this is not a power function.

Question 6

$KE(v) = \frac{1}{2} k v^5$

• Power function check: This is a power function because it has the form $KE(v) = k v^a$, where $k = \frac{1}{2}$ and the exponent $a = 5$.
• Constant of variation: The constant of variation is $\frac{1}{2} k$.

Question 9

$l = d^2$

• Power function check: This is a power function because it has the form $l = kd^a$, where $k = 1$ and the exponent $a = 2$.
• Constant of variation: The constant of variation is $1$.

Question 12

$f(x) = 3x^5$

• Monomial function check: A monomial function has the form $f(x) = ax^n$. This is a monomial function because it follows that form, where $a = 3$ and $n = 5$.
• Degree: The degree is $5$.
• Leading coefficient: The leading coefficient is $3$.

Question 15

$S = 9 \pi r^3$

• Monomial function check: This is a monomial function since it has the form $S = a r^n$, where $a = 9 \pi$ and $n = 3$.
• Degree: The degree is $3$.
• Leading coefficient: The leading coefficient is $9 \pi$.

Question 18

The volume $V$ of a circular cylinder with fixed height is proportional to the square of its radius $r$.

• Power function equation: The relationship described is $V = kr^2$, where $k$ is a constant of variation based on the height of the cylinder.

Question 21

The energy $E$ produced in a nuclear reaction is proportional to the mass $m$, with the constant of variation being $c^2$, the square of the speed of light.

• Power function equation: The equation is $E = c^2 m$.

Would you like more details or have any questions on these solutions?

Here are 5 related questions for further practice:

1. What is the general form of a power function?
2. How can you determine whether an equation is a power function?
3. What is the degree and leading coefficient of $f(x) = 7x^8$?
4. How does the constant of variation affect the behavior of a power function?
5. How would you rewrite $f(x) = x^{\frac{1}{3}}$ as a power function?

Tip: Always check if the exponent is applied to the variable to determine if it's a power function!