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)\\)

Explanation:

Analyze the function transformation

Using the Logarithmic Function Transformations knowledge point

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

The horizontal shift is determined by the term \(x + 3\), which translates the graph \(3\) units to the left.
The vertical shift is determined by the constant term \(+ 2\), which translates the graph \(2\) units up.

Apply transformation to the given point

Using the Logarithmic Function Transformations knowledge point

$$ (x, y) \to (x - 3, y + 2) $$

Applying this rule to the point \((8, 3)\):

$$ (8 - 3, 3 + 2) = (5, 5) $$

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)