5.3 Graphs of Rational Functions of the form f(x) = (ax + b)/(cx + d)
Chapter
Chapter 5
Section
5.3
Purchase this Material for $10
You need to sign up or log in to purchase.
Subscribe for All Access
You need to sign up or log in to purchase.
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}

Buy to View
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.

Buy to View
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.

Buy to View
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}}

Buy to View
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}}

Buy to View
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}}

Buy to View
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}}

Buy to View
1.12mins
Q4d

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

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

Buy to View
2.01mins
Q5a

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

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

Buy to View
2.40mins
Q5b

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

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

Buy to View
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)}}

Buy to View
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

Buy to View
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)
Buy to View
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
Buy to View
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)
Buy to View
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.

Buy to View
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.

Buy to View
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}}.

Buy to View
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?

Buy to View
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?

Buy to View
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?

Buy to View
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.

Buy to View
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?

Buy to View
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?
Buy to View
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?
Buy to View
0.40mins
Q14c

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.

Buy to View
2.47mins
Q14d