Question

exponential functions
equation
$y = 6^{x - 2}+\frac{1}{2}$
domain $(-\infty,\infty)$
range $(\frac{1}{2},\infty)$
x - intercept: none
y - intercept: $(0.5,0)$
increasing $(-\infty,\infty)$
decreasing none
absolute max: none
absolute min: none
end behavior:
as $x\to -\infty,y\to\\_(\frac{1}{2},0)\\_$
as $x\to\infty,y\to\\_\infty\\_$
asymptote
$y = \frac{1}{2}$

Explanation:

Step1: Analyze end - behavior as $x\to-\infty$

For the exponential function $y = 6^{x - 2}+\frac{1}{2}$, as $x\to-\infty$, the term $6^{x - 2}=\frac{6^{x}}{6^{2}}$. Since the base of the exponential function $a = 6>1$, as $x\to-\infty$, $6^{x}\to0$. So $y=6^{x - 2}+\frac{1}{2}\to\frac{1}{2}$.

Step2: Analyze end - behavior as $x\to\infty$

As $x\to\infty$, the term $6^{x - 2}=\frac{6^{x}}{6^{2}}$. Since $a = 6>1$, as $x\to\infty$, $6^{x}\to\infty$, and $y = 6^{x - 2}+\frac{1}{2}\to\infty$.

Answer:

As $x\to-\infty$, $y\to\frac{1}{2}$; as $x\to\infty$, $y\to\infty$