Determine equations for the vertical and horizontal asymptotes of each function.

\displaystyle f(x) = \frac{1}{x-2}

Determine equations for the vertical and horizontal asymptotes of each function.

\displaystyle f(x) = \frac{3}{x+7}

Determine equations for the vertical and horizontal asymptotes of each function.

\displaystyle f(x) = -\frac{4}{x-7}

Determine an equation to represent each function.

6681

Determine an equation to represent each function.

6682

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

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

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

\displaystyle g(x) = -\frac{1}{x-4}

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

\displaystyle h(x) = \frac{1}{2x-3}

Sketch a graph of each function. State the domain, range, y-intercepts, and equations of the asymptotes.

\displaystyle h(x) = - \frac{8}{5x+4}

Determine equations for the vertical asymptotes of the function. Then, state the domain.

\displaystyle f(x) = \frac{1}{(x -3)(x+ 4)}

Determine equations for the vertical asymptotes of the function. Then, state the domain.

\displaystyle f(x) = -\frac{2}{(x+ 3)^2}

Determine equations for the vertical asymptotes of the function. Then, state the domain.

\displaystyle f(x) = \frac{1}{x^2 + 8x + 12}

,For each function,

• i) determine equations for the asymptotes
• ii) determine the y-intercepts
• iii) sketch a graph
• iv) describe the increasing and decreasing intervals
• v) state the domain and range

\displaystyle f(x) = \frac{1}{x^2 + 6x + 5}

,For each function,

• i) determine equations for the asymptotes
• ii) determine the y-intercepts
• iii) sketch a graph
• iv) describe the increasing and decreasing intervals
• v) state the domain and range

\displaystyle f(x) = \frac{1}{x^2 - 5x-24}

,For each function,

• i) determine equations for the asymptotes
• ii) determine the y-intercepts
• iii) sketch a graph
• iv) describe the increasing and decreasing intervals
• v) state the domain and range

\displaystyle f(x) = -\frac{1}{x^2 -6x + 9}

,For each function,

• i) determine equations for the asymptotes
• ii) determine the y-intercepts
• iii) sketch a graph
• iv) describe the increasing and decreasing intervals
• v) state the domain and range

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

Analyse the slope, and change in slope, for the intervals of the function \displaystyle f(x) = \frac{1}{2x^2 +3x - 5} by sketching a graph of the function.

Write an equation for a function that is the reciprocal of a quadratic and has the following properties:

• The horizontal asymptote is y = 0.
• The vertical asymptotes are x = -4 and x = 5
• For the intervals x < -4 and x > 5, y < 0

Determine an equation for the horizontal asymptote of each function.

\displaystyle a(x) = \frac{x}{x + 5}

Determine an equation for the horizontal asymptote of each function.

\displaystyle f(x) = -\frac{2x}{x- 3}

Determine an equation for the horizontal asymptote of each function.

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

Summarize the key features of each function. Then, sketch a graph of the function.

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

Summarize the key features of each function. Then, sketch a graph of the function.

\displaystyle f(x) = - \frac{3x}{x + 1}

Summarize the key features of each function. Then, sketch a graph of the function.

\displaystyle f(x) = \frac{x-2}{x+ 4}

Summarize the key features of each function. Then, sketch a graph of the function.

\displaystyle f(x) = \frac{6x+ 2}{2x-1}

Write an equation of a rational function of the form f(x) = \frac{ax + b}{cx + d} whose graph has all of the following features:

• x-intercept of \frac{1}{4}
• y-intercept of -\frac{1}{2}
• vertical asymptote with equations x = - \frac{2}{3}
• horizontal asymptote with equations x = \frac{4}{3}

Solve algebraically.

\displaystyle \frac{7}{x-4} =2

Solve algebraically.

\displaystyle \frac{3}{x^2 + 6x -24} =1

You can use a graphing device for this. Solve for x.

\displaystyle \frac{x^2 -3x + 1}{2 -x} =\frac{x^2 + 5x + 4}{x-6}

Solve for x.

\displaystyle \frac{3}{x + 5} < 2

Solve for x.

\displaystyle \frac{3}{x + 2} \leq \frac{4}{x + 3}

Solve for x.

\displaystyle \frac{x^2 -x -20}{x^2 -4x -12} > 0

Solve for x.

\displaystyle \frac{x}{x + 5} > \frac{x - 1}{x + 7}

Solve for x.

\displaystyle \frac{x^2 + 5x +4}{x^2 -5x + 6} < 0

Solve for x.

\displaystyle \frac{x^2 -6x + 9}{2x^2 + 17x + 8} > 0

A manufacturer is predicting profit, P, in thousands of dollars. on the sale of x tonnes of fertilizer according to the equation

\displaystyle P(x) = \frac{600x - 15000}{x + 100}

a) Sketch a graph of this relation.

b) Describe the predicted profit as sales increase.

c) Compare the rtes of change of the profit at sales of 100 t and 500 t of fertilizer.

Sketch a graph of each function. Describe each special case.

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

Sketch a graph of each function. Describe each special case.

\displaystyle f(x) = \frac{x^2 -2x - 35}{x^2 -3x - 28}