      UNIT 3  : QUADRATIC FUNCTIONS & EQUATIONS

LESSON 8:  RECIPROCAL FUNCTIONS

Graphing functions and their reciprocals:

Example 1:  x  -4 - 6 -1/6 -2 - 4 - ¼ -1 - 3 -1/3 0 -2 -½ 1 -1 -1 1.5 -0.5 -2 1.75 -0.25 -4 2 0 1/0 = undefined 2.25 0.25 4 2.5 0.5 2 3 1 1 4 2 ½ 5 3 1/3

Please note the following from the graph and table:

·        The graph of y = f(x) =  x – 2 (blue) is a line and its reciprocal (red) has 2 branches separated by the line x = 2 (dashed). It is called a hyperbola.

·        Where f(x) has a zero (x – intercept), the reciprocal has an asymptote (x = 2 dashed)

·        The behaviour near the asymptote is interesting;  as x approaches 2 from the right (x = 3, 2.5, 2.25, 2.1 …), the reciprocal (red) gets very large in the positive direction

·        As x approaches 2 from the left (x = 1, 1.5, 1.75, 1.9 …), the reciprocal (red) gets very large in the negative direction.

·        As x takes on larger positive values, the reciprocal takes on smaller positive values approaching zero from above.

·         As x takes on larger negative values   (-1,-2, -4, -10, …), the reciprocal takes on smaller negative values approaching zero again from below.

·        Where f(x) is positive, the reciprocal is positive;  where f(x) is negative, the reciprocal is negative.

·        Where f(x) = 1, the reciprocal equals 1;  where f(x) = –1 , the reciprocal equals –1. [Points (1, -1),  (3, 1)] Example 2: Solution:

First find the zeros or x – intercepts of y = x + 3.    Let y = 0 and solve for x.

0 = x + 3  and

x = - 3

Hence x = - 3 is the x – intercept of f(x) and the vertical asymptote of the reciprocal.

Find all points on the line where y=1 or y=-1.  [(-2, 1) and (-4, -1)]

Now make a table of values as in above example including values on either side of the asymptote x = -3. x  -11 -8 -1/8 -6 -3 -1/3 -5 -2 -½ -4 -1 -1 -3.5 -0.5 -2 -3.25 -0.25 -4 -3.1 -0.1 -10 -3 0 Undefined -2.9 0.1 10 -2.75 0.25 4 -2.5 0.5 2 -2 1 1 -1 2 ½ 0 3 1/3 1 4 ¼ 8 11 1/11

x =-3

Example 3: Solution:

First find the zeros or x – intercepts of the parabola f(x).  Let y = 0 and solve for x.  Now make a table of values if needed as below including points near the asymptotes x = -2 and x = 4.  See red graph below. x  -4 12 1/12 -3 7 1/7 -2.5 3.25 0.31 -2.1 0.61 1.64 -2.01 0.0601 16.64 -2 0 Undef. -1.99 -0.0599 -16.69 -1.9 -0.59 -1.69 1 -9 -1/9 3.9 -0.59 -1.69 4 0 Undef 4.1 0.61 1.64 5 7 1/7 6 12 1/12

Note values of x taken near asymptotes:

x = - 2.1, -2.01, - 1.9, -1.99, 3.9, 4.1

Please note the following from the graph and table:

·        The graph of y = f(x) =  x2 – 2x - 8 (blue) is a parabola with vertex at (1, -9) and zeros –2, 4. Its reciprocal (red) has 3 branches separated by the lines

x = - 2 and x = 4(dashed).

·        Where f(x) has a zero (x – intercept), the reciprocal has an asymptote (x = - 2 and x = 4 dashed)

·        The behaviour near the asymptotes is interesting;  as x approaches - 2 from the right (x = -1.9, -1.99 in table), the reciprocal (red) gets very large in the negative direction (goes down);  as x approaches - 2 from the left (x = - 2.1, -2.01 in table), the reciprocal (red) gets very large in the positive direction (goes up).  Similar behaviour occurs near the other asymptote x = 4.

·        As x takes on larger positive values (x = 5, 6 in table), the reciprocal takes on smaller values approaching zero from above.  As x takes on larger negative values (- 3, - 4 in table), the reciprocal takes on smaller values approaching zero again from above.

·        Where f(x) is positive, the reciprocal is positive;  where f(x) is negative, the reciprocal is negative.

·        Where f(x) = 1, the reciprocal equals 1;  where f(x) = –1, the reciprocal equals –1. [Points  (+/- 4.2, 1)  and  (+/- 3.8, -1)]. Example 4:

Given the graph of f(x) below, sketch the graph of the reciprocal. Solution:

The zero of f(x) is x = -2.  This is the vertical asymptote of the reciprocal.

Take points on the graph of f(x) and use the mapping below. (-6, 4) ----------------------------à (-6, ¼ )

(-4, 2) ----------------------------à (-4, ½ )

(-3, 1) ----------------------------à (-3, 1)

(-2, 0) ----------------------------à V. Asymp.

(-1, -1) ---------------------------à (-1, -1)

(2, -4) ----------------------------à (2, -¼ )

(4, -6) ----------------------------à (4, -1/6)  Example 5:

Given the graph of f(x) below, sketch the graph of the reciprocal. Solution:

The zeros of f(x) are x = -4 and x = 4.  These are the vertical asymptotes of

the reciprocal. Take points on the graph of f(x) and use the mapping below.   