Question 4.23: Sketch the functions (a) y = 1/x − 0.23 (b) y = 1/x + 3...

(a) The function given in (a) is y = 1/x translated horizontally forward by 0.23 units and (b) is y = 1/x translated vertically up by 3 units.

Step 1: Calculate the value of x that gives rise to division by zero. This will determine the vertical asymptote. Division by zero occurs when the denominator is zero.

x − 0.23 = 0 → x = 0.23

Step 2: Calculate the points of intersection with the y-axis, using the fact that x = 0 on the y-axis:

y=\frac{1}{x-0.23}\rightarrow y=\frac{1}{0-0.23}=-4.35

Step 3: To get some idea of the curvature as the graph approaches the vertical asymptote, calculate the coordinates of some points to its left and right:

See Figure 4.22(a).

(b) The equation of the vertically translated function is

y = \frac{1}{x} + 3

The graph is sketched by translating y = 1/x vertically up the y-axis by 3 units as shown in Figure 4.22(b). The point of intersection with the x-axis is easily found by x solving for when y is zero.