5.3 Graphs of Rational Functions of the form f(x) = (ax + b)/(cx + d)
Chapter
Chapter 5
Section
5.3
Lectures 4 Videos Skechting Hyperbola ex1 Sketching Hyperbola ex2 Increase and Decreasing Intervals for y=ax+bcx+d
Solutions 26 Videos

Match each function with its graph.

 \displaystyle \begin{array}{ccccccc} &a) &h(x)=\displaystyle{\frac{x+4}{2x+5}} &b) & m(x)=\displaystyle{\frac{2x-4}{x-2}} &c) &f(x)=\displaystyle{\frac{3}{x-1}} &d) & g(x)=\displaystyle{\frac{2x-3}{x+2}} \end{array} 2.57mins
Q1

Consider the function f(x)=\displaystyle{\frac{3}{x-2}}.

a) State the equation of the vertical asymptote.

b) State the equation of the horizontal asymptote.

c) Determine the domain and range.

d) Determine the positive and negative intervals.

4.02mins
Q2

Consider the function f(x)=\displaystyle{\frac{4x-3}{x+1}}.

a) State the equation of the vertical asymptote.

b) Determine the behaviour(s) of the function near its vertical asymptote.

c) State the equation of the horizontal asymptote.

d) Use a table of values to determine the end behaviours of the function near its horizontal asymptote.

e) Determine the domain and range.

f) Determine the positive and negative intervals.

3.54mins
Q3

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

y=\displaystyle{\frac{2}{x+3}}

1.45mins
Q4a

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

y=\displaystyle{\frac{x-1}{x-5}}

3.04mins
Q4b

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

y=\displaystyle{\frac{2x+1}{2x-1}}

1.22mins
Q4c

State the equation of the vertical asymptote of each function. Then choose a strategy to determine how the graph of the function approaches its vertical asymptote.

y=\displaystyle{\frac{3x+9}{4x+1}}

1.12mins
Q4d

For the function describe where the function is increasing or decreasing.

f(x)=\displaystyle{\frac{3}{x+5}}

2.01mins
Q5a

For the function describe where the function is increasing or decreasing.

f(x)=\displaystyle{\frac{10}{2x-5}}

2.40mins
Q5b

For the function describe where the function is increasing or decreasing.

f(x)=\displaystyle{\frac{x+5}{4x-1}}

3.13mins
Q5c

For each function, find

• i. determine the domain,
• ii. intercepts, hole
• iii. asymptotes, and
• iv. positive/negative intervals.
• v. region of increasing or decreasing

f(x)=\displaystyle{\frac{x+2}{5(x+2)}}

0.46mins
Q5d

Read each set of conditions. State which of the equation of a rational function of the form f(x)=\displaystyle{\frac{ax+b}{cx+d}} meets these conditions

vertical asymptote at x= -2,

horizontal asymptote at y=0;

negative when x \in (-\infty,-2),

positive when x \in (-2,\infty);

always decreasing

0.50mins
Q6a

Read each set of conditions. State which of the equation of a rational function of the form f(x)=\displaystyle{\frac{ax+b}{cx+d}} meets these conditions.

• vertical asymptote at x = -2,
• horizontal asymptote at y=1;
• x-intercept =0, y-intercept =0;
• positive when x \in (-\infty, -2) or (0,\infty),
• negative when x \in (-2,0)
1.19mins
Q6b

Read each set of conditions. State the equation of a rational function of the form f(x)=\displaystyle{\frac{ax+b}{cx+d}} that meets these conditions.

• hole at x=3; no vertical asymptote; y-intercept =0.5
0.50mins
Q6c

Read each set of conditions. State the equation of a rational function of the form \displaystyle f(x)=\frac{1}{(ax+b)(cx+d)} that meets these conditions. graph.

• vertical asymptotes at x=-2 and x=6,
• horizontal asymptote at y=0;
• positive when x \epsilon (-\infty,-2) or (6,\infty), negative when x \epsilon (-2,6);
• increasing when x \epsilon (-\infty,2),
• decreasing when x \epsilon (2,\infty)
2.58mins
Q6d

Use a graphing calculator to investigate the similarities and differences in the graphs of rational functions of the form f(x)=\displaystyle{\frac{8x}{nx+1}}, for n=1,2,4, and 8.

Find the equation of the horizontal asymptotes as n approach infinity.

1.19mins
Q7ab

For f(x)=\displaystyle{\frac{8x}{nx+1}}, if n is negative, how does the function change as the value of n approaches negative infinity? Choose your own values, and use them as examples to support your conclusions.

0.40mins
Q7c

State the vertical and horizontal asymptote for rational functions f(x)=\displaystyle{\frac{3x+4}{x-1}} and g(x)=\displaystyle{\frac{x-1}{2x+3}}.

2.01mins
Q8

The function I(t)=\displaystyle{\frac{15t+25}{t}} gives the value of an investment, in thousands of dollars, over t years.

a) What is the value of the investment after 2 years?

b) What is the value of the investment after 1 year?

c) What is the value of the investment after 6 months?

1.22mins
Q9abc

The function I(t)=\displaystyle{\frac{15t+25}{t}} gives the value of an investment, in thousands of dollars, over t years.

a) Choose a very small value of t (a value near zero). Calculate the value of the investment at this time. Do you think that the function is accurate at this time? Why or why not?

b) As time passes, what will the value of the investment approach?

1.19mins
Q9ef

An amount of chlorine is added to a swimming pool that contains pure water. The concentration of chlorine, c, in the pool at t hours is given by c(t)=\displaystyle{\frac{2t}{2+t}}, where c is measured in milligrams per litre. What happens to the concentration of chlorine in the pool during the 24 h period after the chlorine is added?

1.21mins
Q10

State

• i. the x-intercept
• ii. the equation and vertical asymptotes
• iii. the equation of horizontal asymptotes

for

f(x)=\displaystyle{\frac{ax+b}{cx+d}} in terms of a, b, c, d.

0.36mins
Q11

Let f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}, g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},, h(x)=\displaystyle{\frac{x^3-3x}{x+1}}, and m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}.

Which of these rational functions has a horizontal asymptote?

0.35mins
Q14a

Let f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}, g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},, h(x)=\displaystyle{\frac{x^3-3x}{x+1}}, and m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}.

• Which has an oblique asymptote?
0.11mins
Q14b

Let f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}, g(x)=\displaystyle{\frac{x^3+8}{x^2+9}},, h(x)=\displaystyle{\frac{x^3-3x}{x+1}}, and m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}.

• Which has no vertical asymptote?
Let f(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}, g(x)=\displaystyle{\frac{x^3+8}{x^2+9}}, h(x)=\displaystyle{\frac{x^3-3x}{x+1}}, and m(x)=\displaystyle{\frac{x^2+x-12}{x^2-4}}.
Graph y= m(x), showing the asymptotes and intercepts.