Question

which best describes the graph of (f(x) = log_2(x + 3) + 2) as a transformation of the graph of (g(x) = log_2 x)?

a translation 3 units right and 2 units up
a translation 3 units left and 2 units up
a translation 3 units up and 2 units right
a translation 3 units up and 2 units left

given that the point ((8, 3)) lies on the graph of (g(x) = log_2 x), which point lies on the graph of (f(x) = log_2(x + 3) + 2)?

((5, 1))
((5, 5))
((11, 1))
((11, 5))

what is the domain of (f(x) = log_2(x + 3) + 2)?

Explanation:

Analyze the transformation of the function

Using the Logarithmic Function Transformations knowledge point

$$ f(x) = \log_2(x + 3) + 2 $$

Comparing this to the parent function \(g(x) = \log_2 x\):

• The term \((x + 3)\) inside the logarithm shifts the graph horizontally by 3 units to the left.
• The term \(+ 2\) outside the logarithm shifts the graph vertically by 2 units up.

Therefore, the transformation is a translation 3 units left and 2 units up.

Determine the transformed point

Using the Logarithmic Function Transformations knowledge point
Given that the point \((8, 3)\) lies on the graph of \(g(x) = \log_2 x\), we apply the horizontal and vertical translations to this point:

• New x-coordinate: \(x' = 8 - 3 = 5\)
• New y-coordinate: \(y' = 3 + 2 = 5\)

Thus, the corresponding point on the graph of \(f(x)\) is \((5, 5)\).

Find the domain of the transformed function

Using the Domain of Logarithmic Functions knowledge point
For the logarithmic function \(f(x) = \log_2(x + 3) + 2\) to be defined, the argument of the logarithm must be strictly positive:

$$ x + 3 > 0 $$

Solving this inequality:

$$ x > -3 $$

In interval notation, the domain is \((-3, \infty)\).

Answer:

Question 1

• a translation 3 units right and 2 units up
• a translation 3 units left and 2 units up (Correct answer)
• a translation 3 units up and 2 units right
• a translation 3 units up and 2 units left

Question 2

• (5, 1)
• (5, 5) (Correct answer)
• (11, 1)
• (11, 5)

Question 3

The domain of \(f(x) = \log_2(x + 3) + 2\) is <blank>\(x > -3\)</blank> (or \((-3, \infty)\)).