Question

Sketch the graph of the exponential equation. $$y=5(0.5)^{x}$$

Short answer

The graph of \(y = 5(0.5)^x\) is a decreasing exponential curve, which starts from the point (0,5), passes through (1, 2.5) and approaches but never reaches the x-axis as x increases.

Step-by-step solution

List the Key Points

To sketch any graph, we first need to identify key points. For exponential functions with base a > 0 and a ≠ 1, y-intercept and an additional point for x = 1 are usually helpful. The y-intercept is obtained when x = 0 which in this case is \(y = 5(0.5)^0 = 5(1) = 5\). Now let's find the value of y when x =1. This gives \(y = 5(0.5)^1 = 5 * 0.5 = 2.5\)

Sketch the Asymptote

Exponential functions also have an asymptote, which is a line that the function approaches but never reaches. The function \(y = 5(0.5)^x\) approaches y = 0 as x moves towards positive infinity. This makes \(y=0\) a horizontal asymptote. Sketch this as a dotted line on your graph.

Plot the Points and Draw the Curve

Plot the points obtained in Step 1 (0,5) and (1,2.5) on the graph. Considering the nature of decreasing exponential function and its asymptotic behavior, the curve should start at the y-intercept and fall towards the asymptote as x increases, never crossing the x-axis. Connect these points with a smooth curve.

Finalize the Graph

Label all important points, the asymptote and the axes of your graph. Your graph depicts the function's behavior and shows all the important components of the exponential function \(y = 5(0.5)^x\).