Question

Sketch the graph of the exponential equation. $$y=0.9^{x}$$

Short answer

The graph of the function \(y=0.9^{x}\) is a downward sloping curve, showing decay, and remains above the x-axis for all x-values.

Step-by-step solution

Behavior of the function

Evaluate the function at some key points to understand its behavior. Consider three points: \(x = -1\), \(x = 0\), and \(x = 1\). Substituting these values into the function gives \(y = 0.9^{-1}\) (for \(x=-1\)), \(y = 0.9^{0}\) (for \(x=0\)), and \(y = 0.9^{1}\) (for \(x=1\)). Thus the points to be plotted are (-1, 1.111), (0, 1), and (1, 0.9)

Draw the plot points

Draw a graph and plot these points that were just obtained: (-1, 1.111), (0, 1), and (1, 0.9)

Sketch the curve

Connect these points with a smooth curve. The curve will descend from left to right, being above the x-axis for all x, and it will be constantly approaching the x-axis as \(x\) increases, but will never reach it. This downward trend indicates the decay of the function.