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}
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.
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.
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}}
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}}
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}}
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}}
For the function describe where the function is increasing or decreasing.
f(x)=\displaystyle{\frac{3}{x+5}}
For the function describe where the function is increasing or decreasing.
f(x)=\displaystyle{\frac{10}{2x-5}}
For the function describe where the function is increasing or decreasing.
f(x)=\displaystyle{\frac{x+5}{4x-1}}
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)}}
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
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)
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
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=-2andx=6, - horizontal asymptote at
y=0; - positive when
x \epsilon (-\infty,-2)or(6,\infty), negative whenx \epsilon (-2,6); - increasing when
x \epsilon (-\infty,2), - decreasing when
x \epsilon (2,\infty)
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.
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.
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}}.
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?
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?
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?
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.
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?
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?
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.