Question

which would be the graph of which of the following is the graph of f (x) = 2^{x - 3} + 4

Explanation:

Step1: Analyze the function \( f(x) = 2^{x - 3}+4 \)

The general form of an exponential function is \( y = a^{x - h}+k \), where \( a>0,a
eq1 \), \( (h,k) \) is the horizontal/vertical shift. For \( f(x)=2^{x - 3}+4 \), \( a = 2>1 \), so it's an increasing exponential function (since the base \( 2>1 \), as \( x \) increases, \( 2^{x-3} \) increases, so \( f(x) \) increases). Also, the horizontal shift is 3 units to the right, and vertical shift is 4 units up. The horizontal asymptote is \( y = 4 \) (as \( x
ightarrow-\infty \), \( 2^{x - 3}
ightarrow0 \), so \( f(x)
ightarrow4 \)).

Step2: Analyze the graphs

• The first, second, and fourth graphs are decreasing (since the curve goes down as \( x \) increases), which can't be for \( f(x)=2^{x - 3}+4 \) (as \( a = 2>1 \), it should be increasing).
• The third graph (orange - colored) is increasing, and we can check the horizontal asymptote. As \( x

ightarrow-\infty \), the function should approach \( y = 4 \). The third graph's curve, as \( x \) decreases, approaches a horizontal line (which should be \( y = 4 \) for our function), and as \( x \) increases, it rises, matching the behavior of an increasing exponential function with \( a = 2>1 \) and vertical shift 4.

Answer:

The graph in the orange - colored card (the third graph from the top) is the graph of \( f(x)=2^{x - 3}+4 \)