# How do you graph y=2xx3^(x-2)-1?

Jun 15, 2017

See explanation

#### Explanation:

$\textcolor{b l u e}{\text{Determine the x-intercept}}$

Set $y = 0 = 2 \left({3}^{x - 2}\right) - 1$

$1 = 2 \left({3}^{x - 2}\right)$

${3}^{x - 2} = \frac{1}{2}$

Lake logs

$\left(x - 2\right) \ln \left(3\right) = \ln \left(\frac{1}{2}\right)$

$x = \frac{\ln \left(1\right) - \ln \left(2\right)}{\ln} \left(3\right) + 2$

$x \approx 1.37$ to 2 decimal places
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$\textcolor{b l u e}{\text{Determine the y-intercept}}$

Set $y = 2 \left({3}^{0 - 2}\right) - 1$

$y = \frac{2}{{3}^{2}} - 1 = - \frac{7}{9}$
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$\textcolor{b l u e}{\text{Determine the extremities}}$

Set $y = k \text{ }$ where $\text{ } k = {\lim}_{x \to + \infty} 2 \left({3}^{x - 2}\right) - 1$

$\text{ } k \to 2 \infty - 1 = \infty$

Thus for $x \to \infty \text{ we have } y \to \infty$

Set $y = k \text{ }$ where $\text{ } k = {\lim}_{x \to - \infty} 2 \left({3}^{x - 2}\right) - 1$

$\text{ } k \to \frac{2}{\infty} - 1 = 0 - 1 = - 1$
This is an asymptote.