Question

use the graph to match each equation with the correct function.
12
$y = 14e^{-0.75x}$
$y = 8e^{-0.2x}$
$y = 8e^{-0.8x}$
$y = 14e^{-0.1x}$

Explanation:

Step1: Analyze the y - intercept

For an exponential function of the form \(y = ae^{bx}\), when \(x = 0\), \(y=a\).

• For \(y = 14e^{-0.75x}\), when \(x = 0\), \(y = 14\).
• For \(y = 8e^{-0.2x}\), when \(x = 0\), \(y = 8\).
• For \(y = 8e^{-0.8x}\), when \(x = 0\), \(y = 8\).
• For \(y = 14e^{-0.1x}\), when \(x = 0\), \(y = 14\).

The functions \(y = 8e^{-0.2x}\) and \(y = 8e^{-0.8x}\) have \(y\) - intercept \(y = 8\), and the functions \(y = 14e^{-0.75x}\) and \(y = 14e^{-0.1x}\) have \(y\) - intercept \(y = 14\).

Step2: Analyze the decay rate

The general form of an exponential decay function is \(y=ae^{bx}\), where \(b<0\). The larger the magnitude of \(b\) (i.e., \(|b|\)), the faster the function decays.

• For \(y = 8e^{-0.2x}\), \(|b|=0.2\).
• For \(y = 8e^{-0.8x}\), \(|b| = 0.8\).
• For \(y = 14e^{-0.75x}\), \(|b|=0.75\).
• For \(y = 14e^{-0.1x}\), \(|b|=0.1\).

Since \(y = 8e^{-0.8x}\) has a larger \(|b|\) among the \(y = 8e^{bx}\) functions, it decays faster. And among the \(y = 14e^{bx}\) functions, \(y = 14e^{-0.75x}\) decays faster.

Answer:

\(y = 14e^{-0.75x}\) \(=\) \(g\)
\(y = 8e^{-0.2x}\) \(=\) \(f\)
\(y = 8e^{-0.8x}\) \(=\) \(j\)
\(y = 14e^{-0.1x}\) \(=\) \(h\)