Question

select the correct answer. which graph represents the function f(x) = 2^{x - 1}+2? a. b. c. d.

Explanation:

Step1: Find the y - intercept

Set \(x = 0\) in \(y=2^{x - 1}+2\). Then \(y=2^{0 - 1}+2=2^{-1}+2=\frac{1}{2}+2=\frac{1 + 4}{2}=\frac{5}{2}=2.5\).

Step2: Analyze the horizontal asymptote

The general form of an exponential function is \(y = a\cdot b^{x - h}+k\). For \(y = 2^{x - 1}+2\), as \(x\to-\infty\), \(2^{x - 1}\to0\), so \(y\to2\). The horizontal asymptote is \(y = 2\).

Step3: Analyze the behavior as \(x\to+\infty\)

As \(x\to+\infty\), since the base \(b = 2>1\) in the exponential function \(2^{x - 1}\), \(y = 2^{x - 1}+2\to+\infty\).

Looking at the graphs:

• Option A: The horizontal asymptote is \(y = 2\) and the y - intercept is around \(2.5\) and it increases as \(x\) increases.
• Option B: The y - intercept is around \(4\), so it's incorrect.
• Option C: The horizontal asymptote is not \(y = 2\), so it's incorrect.
• Option D: The function is decreasing as \(x\) increases, while our function \(y = 2^{x - 1}+2\) should increase as \(x\) increases since the base of the exponential part is \(2>1\), so it's incorrect.

Answer:

A.