Question

use the graph to match each equation with the correct function.

$y = 2e^{-0.1x}$

$y = 4e^{-0.1x}$

$y = 4e^{-1.5x}$

$y = 2e^{-1.5x}$

Explanation:

Step1: Analyze the general form of exponential functions

The general form of an exponential function is \(y = ae^{bx}\). When \(b<0\), it is a decaying exponential function. The larger the absolute value of \(b\), the faster the decay. When \(x = 0\), \(y=a\) (the \(y -\)intercept).

Step2: Find the \(y -\)intercepts

For \(y = 2e^{-0.1x}\), when \(x = 0\), \(y=2\). For \(y = 4e^{-0.1x}\), when \(x = 0\), \(y = 4\). For \(y=4e^{-1.5x}\), when \(x = 0\), \(y = 4\). For \(y = 2e^{-1.5x}\), when \(x = 0\), \(y=2\).

Step3: Compare the decay rates

Since \(\vert- 1.5\vert>\vert-0.1\vert\), the functions \(y = 4e^{-1.5x}\) and \(y = 2e^{-1.5x}\) decay faster than \(y = 4e^{-0.1x}\) and \(y = 2e^{-0.1x}\).

Answer:

\(y = 2e^{-0.1x}\) matches \(f\), \(y = 4e^{-0.1x}\) matches \(h\), \(y = 4e^{-1.5x}\) matches \(g\), \(y = 2e^{-1.5x}\) matches \(j\)