Question

enter the equation \\(g(x) = \frac{1}{4} \cdot 2^{x+1} - 5\\) and \\(f(x) = 2^x\\).
how does the graph g(x) differ from the graph of f(x)? select all that apply.

the graph translates down 5 units.
the graph compressed vertically by a scale factor of \\(\frac{1}{4}\\)
the graph translates left 1 units
the graph compressed horizontally by a scale factor of \\(\frac{1}{4}\\)
the graph reflected across the x-axis

Explanation:

Analyze the vertical translation

$$ g(x) = \frac{1}{4} \cdot 2^{x+1} - 5 $$

The constant term \(-5\) subtracted at the end of the function shifts the entire graph vertically downward by \(5\) units.

Analyze the vertical compression

$$ a = \frac{1}{4} $$

Multiplying the base function by a factor of \(a\) where \(0 < a < 1\) results in a vertical compression. Here, the factor is \(\frac{1}{4}\), which compresses the graph vertically by a scale factor of \(\frac{1}{4}\).

Analyze the horizontal translation

$$ 2^{x+1} = 2^{x - (-1)} $$

Replacing \(x\) with \(x + 1\) in the exponent shifts the graph horizontally to the left by \(1\) unit.

Answer:

• The graph translates down 5 units. (Correct answer)
• The graph compressed vertically by a scale factor of \(\frac{1}{4}\) (Correct answer)
• The graph translates left 1 units (Correct answer)
• The graph compressed horizontally by a scale factor of \(\frac{1}{4}\)
• The graph reflected across the x-axis