Question

graphing a logarithmic function after a transformation

the graph of \\(f(x) = \log x + 3\\) is the graph of \\(g(x) = \log x\\) translated 3 units up.

if you are given the graph of \\(g\\), how could you graph \\(f(x) = \log x + 3\\)?
for each point on the graph of \\(g\\), add 3 to the y-coordinate.

which is the graph of \\(f(x) = \log x + 3\\)?

Explanation:

Analyze the transformation

Using the Logarithmic Function Transformations knowledge point

$$ f(x) = \log x + 3 = g(x) + 3 $$

This represents a vertical translation of the parent graph \(g(x) = \log x\) upward by 3 units.

Evaluate coordinates of the parent function

Using the Logarithmic Functions knowledge point

$$
LATEXBLOCK0
$$

Determine coordinates of the transformed function

Using the Logarithmic Function Transformations knowledge point

$$
LATEXBLOCK1
$$

Identify the correct graph

Using the Logarithmic Function Transformations knowledge point

• The parent graph \(g(x)\) (orange curve) passes through \((1, 0)\).
• The transformed graph \(f(x)\) (blue curve) must pass through \((1, 3)\).
• In the first option, the blue curve \(f\) passes through \((1, 3)\) (exactly 3 units above the x-axis at \(x = 1\)).
• In the second option, the blue curve \(f\) passes through \((1, 2)\), which is a vertical shift of only 2 units.
• Therefore, the first graph is correct.

Answer:

• (A) The first graph, where the blue curve \(f(x)\) is shifted 3 units up from the orange curve \(g(x)\) and passes through \((1, 3)\). (Correct answer)
• (B) The second graph, where the blue curve \(f(x)\) is shifted 2 units up from the orange curve \(g(x)\) and passes through \((1, 2)\).