Questions: Graph the exponential function. f(x)=-2(4/5)^x

Transcript text: Graph the exponential function. \[ f(x)=-2\left(\frac{4}{5}\right)^{x} \]

Solution

Solution Steps

Step 1: Calculate five points

We need to find five points for the function $f(x) = -2(\frac{4}{5})^x$. Let's choose $x = -2, -1, 0, 1,$ and $2$.

$f(-2) = -2(\frac{4}{5})^{-2} = -2(\frac{5}{4})^2 = -2(\frac{25}{16}) = -\frac{25}{8} = -3.125$
$f(-1) = -2(\frac{4}{5})^{-1} = -2(\frac{5}{4}) = -\frac{5}{2} = -2.5$
$f(0) = -2(\frac{4}{5})^0 = -2(1) = -2$
$f(1) = -2(\frac{4}{5})^1 = -2(\frac{4}{5}) = -\frac{8}{5} = -1.6$
$f(2) = -2(\frac{4}{5})^2 = -2(\frac{16}{25}) = -\frac{32}{25} = -1.28$

So, the five points are $(-2, -3.125)$, $(-1, -2.5)$, $(0, -2)$, $(1, -1.6)$, and $(2, -1.28)$.

Step 2: Plot the points

Plot the five points calculated in the previous step on the graph.

Step 3: Draw the graph

Connect the points with a smooth curve. The graph should approach the x-axis but never touch it. The horizontal asymptote is $y=0$.

Final Answer:

The five points are $(-2, -3.125)$, $(-1, -2.5)$, $(0, -2)$, $(1, -1.6)$, and $(2, -1.28)$. Plot these points and connect them with a smooth curve that approaches the x-axis as x increases. The horizontal asymptote is $y=0$.

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