Questions: Find an equation in the form of y=f(x) for the graph sketched below. Use base 2 for any logarithmic or exponential functions.

Solution Steps

The graph appears to be an exponential function because it shows rapid growth as \( x \) increases and approaches a horizontal asymptote as \( x \) decreases.

The horizontal asymptote is at \( y = 5 \). This suggests that the function has been shifted vertically by 5 units.

A point on the graph is \((0, 6)\). This point will help us determine the specific parameters of the exponential function.

The general form of an exponential function with a horizontal asymptote at \( y = 5 \) is: \[ y = a \cdot 2^{bx} + 5 \]

Using the point \((0, 6)\): \[ 6 = a \cdot 2^{b \cdot 0} + 5 \] \[ 6 = a \cdot 1 + 5 \] \[ a = 1 \]

Since \( a = 1 \), the equation simplifies to: \[ y = 2^{bx} + 5 \]

To find \( b \), we need another point. However, since we only have one point, we assume \( b = 1 \) for simplicity, which fits the general shape of the graph.

Final Answer