Question

graph the function ( g(x) = -5^{x} ) and its asymptote. graph the asymptote as a dashed line. use the graphing tool to graph the function.

Explanation:

Step1: Analyze the general form of exponential function

The general form of an exponential function is \(y = a\cdot b^{x}+k\). For the function \(g(x)=-5^{x}\), we can rewrite it as \(g(x)=-1\cdot5^{x}+0\). Here \(a=- 1\), \(b = 5\), \(k = 0\).

Step2: Determine the horizontal asymptote

For an exponential function of the form \(y=a\cdot b^{x}+k\), the horizontal asymptote is \(y = k\). Since \(k = 0\) in the function \(g(x)=-5^{x}\), the horizontal asymptote is \(y = 0\).

Step3: Find some key points

When \(x = 0\), \(g(0)=-5^{0}=-1\).
When \(x = 1\), \(g(1)=-5^{1}=-5\).
When \(x=-1\), \(g(-1)=-5^{-1}=-\frac{1}{5}=-0.2\).

Answer:

Plot the points \((0, - 1)\), \((1,-5)\), \((-1,-0.2)\) and draw a curve passing through these points. Draw the horizontal asymptote \(y = 0\) as a dashed line.