Question

Sketch the graph of the exponential equation. $$y=\frac{1}{2}(2)^{x}$$

Short answer

The graph of exponential function \(y=\frac{1}{2}(2)^{x}\) starts at the y-intercept (0,1), and is an upward curve which gradually rises from left to right on the coordinate plane.

Step-by-step solution

Understand the Exponential Function

In the given equation, \(y=\frac{1}{2}(2)^{x}\), the base of the exponent is 2, and the function is multiplied by \(\frac{1}{2}\). This is an exponential function, which means that the graph would start from a certain point on the y-axis and increase as x increases when the base is greater than 1.

Identify Y-Intercept

To find the y-intercept of an exponential function, plug in 0 for x. So, \(y=\frac{1}{2}(2)^{0}\) gives y = 1. Hence, the y-intercept of the function is at (0,1).

Identify Shape of Graph

An exponential function graph with a base greater than 1 rises from left to right on the coordinate plane, which means it is an increasing function. It might be useful to plot a few more points to get a better idea about the shape of the graph. For example, for x=1, \(y=\frac{1}{2}(2)^{1}=1\). For x=2, \(y=\frac{1}{2}(2)^{2}=2\). For x=-1, \(y=\frac{1}{2}(2)^{-1}=0.25\). Given these points and the nature of an exponential function, we can sketch a curve that rises from left to right starting at the y-intercept.

Sketch the Graph

Having the y-intercept at point (0,1), and some other points for guidance like (1, 1), (2, 2), and (-1, 0.25), plot these points on the coordinate plane. Connect these points carefully remembering that the curve should rise from left to right. This curve represents the graph of the given exponential function \(y=\frac{1}{2}(2)^{x}\).