Question

let the graph of g be a reflection in the x - axis followed by a translation 2 units down of the graph of f(x)=e^{x}. write a rule for g. g(x)=\square

Explanation:

Step1: Reflect the function in the x - axis

When a function \(y = f(x)\) is reflected in the \(x\) - axis, the transformation rule is \(y=-f(x)\). Given \(f(x)=e^{x}\), after reflection in the \(x\) - axis, the function becomes \(y = -e^{x}\).

Step2: Translate the function 2 units down

When a function \(y = h(x)\) is translated \(k\) units down (\(k>0\)), the transformation rule is \(y=h(x)-k\). Here \(h(x)=-e^{x}\) and \(k = 2\). So, \(g(x)=-e^{x}-2\).

Answer:

\(g(x)=-e^{x}-2\)