Question

Sketch the graph of the function. $$ y=e^{-x / 4} $$

Short answer

The graph of the function \(y=e^{-x / 4}\) is an exponential decay curve. It starts from a high y-value (infinity) at negative infinity x and decays towards the x-axis without ever reaching it as \(x\) increases.

Step-by-step solution

Construct a Table of Values

Choose a variety of x-values and calculate their corresponding y-values using the function \(y=e^{-x / 4}\). For example, when \(x=0\), \(y=e^{-0 / 4}=1\). When \(x=4\), \(y=e^{-4 / 4}=e^{-1}\). Repeat this process for a range of negative and positive x-values.

Plot the Points

Once you have a list of (x, y) pairs, plot them on a set of axes. Include negative x-values to show the behavior of the function. Ensure to label each point appropriately.

Draw the Curve

Connect the plotted points with a smooth curve. Note that the function approaches but never reaches the x-axis as \(x\) increases and the function approaches infinity as \(x\) decreases.