      UNIT 3  : QUADRATIC FUNCTIONS & EQUATIONS

LESSON 8:  RECIPROCAL FUNCTIONS HOMEWORK QUESTIONS

Quick Review : 1.  For each of the following, sketch the graph and its reciprocal.   2. Given the function y = x  – 2.

a)  Sketch the graph and its reciprocal.

b)  State the domain and range of the reciprocal.

c)  Find the equation of the inverse of the reciprocal function.

3.  Sketch the graph of the reciprocal function for each of the following.

a) b) c) Solutions:

1.  For each of the following, sketch the graph and its reciprocal. Solution:

First find the zeros or x – intercepts of y = 2x - 3.    Let y = 0 and solve for x.

0 = 2x - 3  and

2x = 3

x = 1.5

Hence x = 1.5 is the x – intercept of f(x) and the vertical asymptote (VA) of the reciprocal.

Now graph the straight line using the x-intercept and points in table below. (blue graph below)

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

Now make a table of values including points near the asymptote  x = 1.5. x  -3 -9 -1/9= -0.11 -2 -7 - 1/7= -0.14 -1 -5 -1/5= -0.2 0 -3 -1/3= -0.33 1 -1 -1 1.4 -0.2 -1/0.2= -5 1.5 0 VA 1.6 0.2 5 2 1 1 3 3 0.33 4 5 0.2 5 7 0.14

Note values of x taken near asymptote:

x =1.4, 1.6

See graph at left (red).  Solution:

First find the zeros or x – intercepts.  Let y = 0 and solve for x. Next find the vertex of the parabola f(x):. Now graph the parabola using the x-intercepts and vertex calculated above. (blue graph below)

Find all points on the parabola where y=1 or y=-1.  [approx.  (3.05, 1), (-3.05, 1), and (2.95, -1), (-2.95, -1)] Now make a table of values if needed including points near the asymptotes x = -3 and x = 3. x  -5 16 1/16 -4 7 1/7 -3.5 3.25 0.31 -3.1 0.61 1.64 -3.01 0.0601 16.64 -3 0 Undef - VA -2.99 -0.0599 -16.69 -2.9 -0.59 -1.69 0 -9 -1/9 2.9 -0.59 -1.69 3 0 Undef - VA 3.1 0.61 1.64 4 7 1/7 5 16 1/16

Note values of x taken near asymptotes:

x = - 3.1, -3.01, - 2.9, -2.99, 2.9, 3.1

Please note the following from the graph and table:

·        The graph of y = f(x) =  x2 - 9 (blue) is a parabola with vertex at (0, -9) and zeros –3, 3. Its reciprocal (red) has 3 branches separated by the lines

x = - 3 and x = 3(dashed).

·        Where f(x) has a zero (x – intercept), the reciprocal has an asymptote (x = - 3 and x = 3 (dashed))

·        The behaviour near the asymptotes is interesting;  as x approaches - 3 from the right (x = -2.9, -2.99 in table), the reciprocal (red) gets very large in the negative direction (goes down);  as x approaches - 3 from the left (x = - 3.1, -3.01 in table), the reciprocal (red) gets very large in the positive direction (goes up).  Similar behaviour occurs near the other asymptote x = 3.

·        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. Solution:

First find the zeros or x – intercepts.  Let y = 0 and solve for x. Next find the vertex. Now graph the parabola using the x-intercepts and vertex calculated above. (blue graph below)

Find all points on the parabola where y=1 or y=-1. approx.  [(1.2, 1), (-3.2, 1), and (0.7, -1), (-2.7, -1)] Now make a table of values if needed including points near the asymptotes x = -3 and x = 1 if needed. x  -5 12 1/12 -4 5 1/5 -3.5 2.25 1/2.25 = 0.44 -3.1 0.41 2.44 -3.01 0.0401 24.94 -3 0 Undef.(VA) -2.99 -0.0399 -25.06 -2.9 -0.39 -2.56 -2 -3 -1/3 -1 -4 -¼ 0 -3 -1/3 0.9 -0.39 -2.56 1 0 Undef.(VA) 1.1 0.41 2.44 2 5 1/5 3 12 1/12

Note values of x taken near asymptotes:

x = - 3.1, -3.01, - 2.9, -2.99, 0.9, 1.1

Please note the following from the graph and table:

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

·        Where f(x) has a zero (x – intercept), the reciprocal has an asymptote (x = - 3 and x = 1 dashed)

·        As x approaches - 3 from the right (x = -2.9, -2.99 in table), the reciprocal (red) gets very large in the negative direction (goes down).

·        As x approaches - 3 from the left (x = - 3.1, -3.01 in table), the reciprocal (red) gets very large in the positive direction (goes up).

·        Similar behaviour occurs near the other asymptote x = 1.

·        As x takes on larger positive values (x = 2, 3 in table), the reciprocal takes on smaller values approaching zero from above.

·        As x takes on larger negative values (- 4, - 5 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.

2. Given the function y = x  – 2.

a)  Sketch the graph and its reciprocal.

b)  State the domain and range of the reciprocal.

c)  Find the equation of the inverse of the reciprocal function.

Solution:

First find the zeros or x – intercepts of y = x – 2.    Let y = 0 and solve for x.

0 = x –2  and

x = 2

Hence x = 2 is the x – intercept of f(x) and the vertical asymptote of the reciprocal.

Now make a table of values as in above example including values on either side of the asymptote x = 2. x  -6 -8 -1/8 -1 -3 -1/3 0 -2 -½ 1 -1 -1 1.5 -0.5 -2 1.9 -0.1 -10 1.99 -0.01 -100 2 0 Undefined 2.1 0.1 10 2.01 0.01 100 2.5 0.5 2 3 1 1 4 2 ½ 5 3 1/3 6 4 ¼

Note values of x taken near asymptotes:

x = 1.9, 1.99, 2.1, 2.01

b)  For the domain,  the graph extends indefinitely to the left and right but does not exist at x = 2.  Therefore For the range,  the graph extends indefinitely up and down but does not exist at y = 0.  Therefore c)  For the equation of the inverse, interchange x and y and then isolate y. 3.  Sketch the graph of the reciprocal function for each of the following.

a) Solution:

The zero of f(x) is x = 1.  This is the vertical asymptote of the reciprocal.

Take points on the graph of f(x) and use the mapping below. (-3, 4) ----------------------------à (-3, ¼ )

(-1, 2) ----------------------------à (-1, ½ )

(0, 1) ----------------------------à (0, 1)

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

(2, -1) ---------------------------à (2, -1)

(5, -4) ----------------------------à (5, -¼ )

See graph below (red). b) Solution:

The zeros of f(x) are x = - 9  and  x = 9.  These are the vertical asymptotes of the reciprocal.

Take points on the graph of f(x) and use the mapping below. See graph below (red). Note that where f(x) = 1, the reciprocal function (red) equals 1

– ie – they meet where y = 1 on the graph  c) 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. (-4, -3) ----------------------------à (-4, -1/3 )

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

(0, -2) -----------------------------à (0, -½ )

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

(1.5, -½ ) --------------------------à (1.5, -2)

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

(2.5, -½ ) --------------------------à (2.5, -2)

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

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

See graph below (red). 