Explore: Graphing Reciprocal Functions Solutions

Graphing Reciprocal Functions Solutions

1. 1. Find the values of a, h, and k . 
  2. Plot the asymptotes and any intercepts on the graph. 
  3. Sketch the hyperbola using the value of a to determine placement and stretch or compression .

Be sure to label the key details of the graph because the coordinate planes are unscaled.
The table of the parent function shows that many of the values are fractional .
Therefore, when graphing a reciprocal function, sketch the hyperbola rather than plot points.

Graph the asymptotes and intercepts. Sketch the hyperbola.

$f (x) = - \frac{1}{x} + 3$

Plan 
Name a, h, k 

Graph asymptotes 

Solve for any intercepts 

Graph intercepts 

Sketch hyperbola

Implement
$a = - 1, h = 0, k = 3$

Intercepts:

$\begin{array}{rcl} 0 & = & - \frac{1}{x} + 3 \\ - 3 & = & - \frac{1}{x} \\ - 3 x & = & - 1 \\ x & = & \frac{1}{3} \end{array}$

no y-intercept

Explain

The hyperbola is a reflection because a = –1. The graph will shift up 3 spaces from the parent graph.

Note

You should use a, h, and k to describe the graph (written or verbal) before sketching so you know what to expect from the graph.

Graph the asymptotes and intercepts. Sketch the hyperbola.

$g (x) = \frac{2}{x - 4} - 2$

$a = 2, h = 4, k = - 2$

Intercepts:

$\begin{array}{rcl} 0 & = & \frac{2}{x - 4} - 2 \\ 2 & = & \frac{2}{x - 4} \\ 2 (x - 4) & = & 2 \\ x - 4 & = & 1 \\ x & = & 5 \\ (5, 0) \end{array}$

$g (0) = \frac{2}{0 - 4} - 2 g (0) = - \frac{1}{2} - 2 g (0) = - 2.5 (0, - 2.5)$

The hyperbola stretched, making it farther away from the asymptotes when a = 2. The values of h and k shift the function right 4, and down 2.

Graph the asymptotes and intercepts. Sketch the hyperbola.

$f (x) = \frac{1}{3 (x - 1)}$

$a = \frac{1}{3}, h = 1, k = 0$

Intercepts:

no x-intercept

$f (0) = \frac{1}{3 (0 - 1)} f (0) = \frac{1}{- 3} (0, - \frac{1}{3})$

The hyperbola is closer to the asymptotes since the value of a is between 0 and 1. The graph shifts right one space.