Question

the graphs of (f(x) = 10^x) and its translation, (g(x)), are shown.
what is the equation of (g(x))?

(g(x) = 10^{x-3})
(g(x) = 10^{x+3})
(g(x) = 10^x - 3)
(g(x) = 10^x + 3)

Explanation:

Identify corresponding points on both graphs

Compare the labeled points on the graph of \(f(x) = 10^x\) and its translation \(g(x)\):

• \((0, 1) \to (3, 1)\)
• \((1, 10) \to (4, 10)\)
• \((2, 100) \to (5, 100)\)

Determine the horizontal shift

Calculate the change in the \(x\)-coordinates for each corresponding pair:

$$ \Delta x = 3 - 0 = 3 $$

$$ \Delta x = 4 - 1 = 3 $$

$$ \Delta x = 5 - 2 = 3 $$

The graph is translated horizontally to the right by \(3\) units.

Write the equation for the translated function

A horizontal shift of a function \(f(x)\) to the right by \(h\) units is represented by \(f(x - h)\):

$$ g(x) = f(x - 3) = 10^{x - 3} $$

Answer:

• (A) \(g(x) = 10^{x - 3}\) (Correct answer)
• (B) \(g(x) = 10^{x + 3}\)
• (C) \(g(x) = 10^x - 3\)
• (D) \(g(x) = 10^x + 3\)