Question

use the interactive graph below to sketch a graph of $y = 3^{4 - x} + 1$. place the asymptote before placing the two points.

Explanation:

Step1: Find horizontal asymptote

For $y = 3^{4-x} + 1$, as $x\to\infty$, $3^{4-x}=\frac{3^4}{3^x}\to0$. So asymptote: $y=1$

Step2: Calculate first point (x=4)

Substitute $x=4$:
$y=3^{4-4}+1=3^0+1=1+1=2$
Point: $(4, 2)$

Step3: Calculate second point (x=3)

Substitute $x=3$:
$y=3^{4-3}+1=3^1+1=3+1=4$
Point: $(3, 4)$

Step4: Calculate third point (x=5)

Substitute $x=5$:
$y=3^{4-5}+1=3^{-1}+1=\frac{1}{3}+1=\frac{4}{3}\approx1.33$
Point: $(5, \frac{4}{3})$

Answer:

1. Draw the horizontal asymptote $y=1$ (a dashed horizontal line through $y=1$)
2. Plot the points $(4, 2)$, $(3, 4)$, and $(5, \frac{4}{3})$
3. Draw a curve that approaches $y=1$ as $x\to\infty$, passes through the plotted points, and increases as $x$ decreases.