Let's go through the questions and solve them one by one.

Question 27:

The transformation is from $f(x) = -6|x + 2| - 2$ to $g(x) = -6|x + 4| + 5$.

The general form of a transformation for absolute value functions is: $f(x) = a|x - h| + k$

Where:

• $h$ is the horizontal shift.
• $k$ is the vertical shift.

For $f(x) = -6|x + 2| - 2$:

• The graph is shifted 2 units to the left (because $+2$) and 2 units downward (because of the $-2$).

For $g(x) = -6|x + 4| + 5$:

• The graph is shifted 4 units to the left (because $+4$) and 5 units upward (because of the $+5$).

Thus, the transformation is:

• Horizontal translation of 2 units to the left and vertical translation of 7 units upward.

Answer: (c)

Question 28:

The function increases by a factor of 4 over every unit interval in $x$, and $f(0) = 1$.

An exponential function that grows by a factor of 4 is: $f(x) = 4^x$

Also, $f(0) = 1$, which confirms that: $f(0) = 4^0 = 1$

Thus, the function rule is: Answer: (b) $f(x) = 4^x$

Question 29:

The circumference of a circle increases from $\pi$ cm to $2\pi$ cm.

The formula for the circumference of a circle is: $C = 2\pi r$

If the circumference doubles, then the radius also doubles. Since the area of a circle is: $A = \pi r^2$

If $r$ doubles, then the area becomes $4$ times the original area (because $(2r)^2 = 4r^2$).

Answer: (d) It is quadrupled

Question 30:

We are given a right triangle where the hypotenuse is 13, and one leg is 12. Using the Pythagorean Theorem: $a^2 + b^2 = c^2$

Where:

• $a = 5$ (the other leg),
• $b = 12$,
• $c = 13$.

The missing side $AB$ is 5 units.

Answer: (a) 5

Do you want more details on any of these problems?

Here are 5 related questions to expand your understanding:

1. How do you graph transformations of absolute value functions step by step?
2. What is the difference between linear and exponential growth?
3. Why does doubling the radius of a circle quadruple its area?
4. Can the Pythagorean theorem be used for triangles that are not right-angled?
5. How do transformations affect the domain and range of a function?

Tip: In transformation problems, always track how shifts, stretches, and reflections affect the graph's shape and position.