Question

1. $k(x) = \frac{1}{x - 3} + 5$

Explanation:

Step1: Identify Parent Function

The parent function is $y = \frac{1}{x}$, a hyperbola.

Step2: Determine Transformations

• Horizontal shift: The term $x - 3$ shifts the graph 3 units to the right (since $x - h$ shifts right by $h$).
• Vertical shift: The $+5$ shifts the graph 5 units up.

Step3: Find Asymptotes

• Vertical asymptote: From $x - 3 = 0$, so $x = 3$.
• Horizontal asymptote: From the vertical shift, $y = 5$.

Step4: Plot Key Points

• For the parent $y = \frac{1}{x}$, points like $(1,1)$, $(-1,-1)$ exist. After shifting:
• $(1 + 3, 1 + 5) = (4,6)$
• $(-1 + 3, -1 + 5) = (2,4)$
• Also, test $x = 4$: $k(4) = \frac{1}{1} + 5 = 6$; $x = 2$: $k(2) = \frac{1}{-1} + 5 = 4$; $x = 5$: $k(5) = \frac{1}{2} + 5 = 5.5$; $x = 1$: $k(1) = \frac{1}{-2} + 5 = 4.5$.

Step5: Sketch the Graph

Draw the two branches of the hyperbola, approaching the asymptotes $x = 3$ and $y = 5$, passing through the plotted points.

Answer:

To graph $k(x) = \frac{1}{x - 3} + 5$:

1. Asymptotes: Vertical at $x = 3$, horizontal at $y = 5$.
2. Transformations: Shift $y = \frac{1}{x}$ 3 units right, 5 units up.
3. Key Points: Plot $(4,6)$, $(2,4)$, $(5, 5.5)$, $(1, 4.5)$, etc., then draw the hyperbola branches approaching the asymptotes.