2.05 Transformations of hyperbolas
Consider the function y = - \dfrac{1}{4 x}.
Complete the following table of values:
| x | -3 | -2 | -1 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
Sketch the graph.
In which quadrants does the graph lie?
Consider the function y = \dfrac{2}{x + 4}.
State the equation of the vertical asymptote.
State the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the hyperbolic function y = \dfrac{3}{x} - 3.
Determine whether the following graphs indicate the position of the hyperbola's branches relative to its asymptotes:
Which curve approaches positive and negative infinity more quickly: y = \dfrac{1}{x} or y = \dfrac{3}{x} - 3?
What are the equations of the vertical and horizontal asymptotes of y = \dfrac{3}{x} - 3?
Sketch the graph of y = \dfrac{3}{x} - 3.
Consider the hyperbolic function y = -\dfrac{3}{x} + 2.
Determine whether the following graphs indicate the position of the hyperbola's branches relative to its asymptotes:
Which curve approaches positive and negative infinity more quickly: y = \dfrac{- 1}{x} or
y = \dfrac{- 3}{x} + 2
What are the equations of the vertical and horizontal asymptotes of y = \dfrac{- 3}{x} + 2?
Sketch the graph of y = \dfrac{- 3}{x} + 2.
Consider the hyperbolic function y = \dfrac{1}{5 x} - 3.
Determine whether the following graphs indicate the position of the hyperbola's branches relative to its asymptotes:
Which curve approaches positive and negative infinity more quickly: y = \dfrac{1}{x} or
y = \dfrac{1}{5 x} - 3
What are the equations of the vertical and horizontal asymptotes of y = \dfrac{1}{5 x} - 3?
Sketch the graph of y = \dfrac{1}{5 x} - 3.
Consider the hyperbolic function y = - \dfrac{1}{2 x} + 2.
Determine whether the following graphs indicate the position of the hyperbola's branches relative to its asymptotes:
Which of these curves approach positive and negative infinity more quickly:
y = \dfrac{- 1}{x} or y = - \dfrac{1}{2 x} + 2
What are the equations of the vertical and horizontal asymptotes of y = - \dfrac{1}{2 x} + 2?
Sketch the graph of y = - \dfrac{1}{2 x} + 2.
Consider the function y = \dfrac{1}{x - 3}.
Complete the table of values:
| x | 1 | 2 | \dfrac{5}{2} | \dfrac{7}{2} | 4 | 5 |
|---|---|---|---|---|---|---|
| y |
Sketch the graph.
For each of the following hyperbolic functions:
Determine which of the following graphs indicate the position of the hyperbola's branches relative to its asymptotes: A or B.
Write down the equations of the vertical and horizontal asymptotes.
Sketch the graph.
y = \dfrac{1}{4 \left(x - 2\right)}
y = \dfrac{- 2}{x - 1}
y = - \dfrac{1}{2 \left(x - 2\right)} + 2
Consider the function y = \dfrac{x - 2}{x - 4}.
Solve the following equation for a:\dfrac{x - 2}{x - 4} = \dfrac{x - 4 + a}{x - 4}
Hence, express y = \dfrac{x - 2}{x - 4} in the form y = \dfrac{m}{x - h} + k, for some values k and h.
State the equation of the vertical asymptote.
As x approaches \infty, what does y approach?
Hence, state the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the function y = \dfrac{x - 4}{x - 3}.
Solve the following equation for a:\dfrac{x - 4}{x - 3} = \dfrac{x - 3 - a}{x - 3}
Hence, express y = \dfrac{x - 4}{x - 3} in the form y = \dfrac{k}{x - 3} + h, for some values k and h.
State the equation of the vertical asymptote.
As x approaches \infty, what does y approach?
Hence, state the equation of the horizontal asymptote.
Sketch the graph of the function
Consider the function y = \dfrac{4 - x}{x - 3}.
Express y = \dfrac{4 - x}{x - 3} in the form y = \dfrac{k}{x - 3} + h, for some values k and h.
State the equation of the vertical asymptote.
As x approaches \infty, what does y approach?
Hence, state the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the graph of the hyperbola y = \dfrac{1}{x}:
What would be the new equation if the graph was shifted upwards by 4 units?
What would be the new equation if the graph was shifted to the right by 7 units?
Consider the following hyperbolas:
y = \dfrac{6}{x} \text{ and } y = \dfrac{6}{x} - 3What is the y value of y = \dfrac{6}{x} corresponding to x = - 2 ?
What is the y value of y = \dfrac{6}{x} - 3 corresponding to x = - 2 ?
How is y = \dfrac{6}{x} transformed to make y = \dfrac{6}{x} - 3 ?
Consider the following hyperbolas:
y = \dfrac{- 1}{x} \text{ and } y = \dfrac{- 1}{x - 4}What value cannot be substituted for x in y = \dfrac{-1}{x} ?
In which quadrants does y = \dfrac{-1}{x} lie?
What value cannot be substituted for x in y = \dfrac{-1}{x-4} ?
In which quadrants does y = \dfrac{-1}{x-4} lie?
How can the graph of y = \dfrac{-1}{x} be transformed to create the graph of y = \dfrac{-1}{x-4} ?
A hyperbola has a domain of x \in \Reals, x \neq 2 and a range of y \in \Reals, y \neq - 3.
Determine whether the following could be the equation of the hyperbola:
y = \dfrac{1}{x - 2} - 3
y = \dfrac{3}{x - 2} + 3
y = \dfrac{1}{3 \left(x - 2\right)} - 3
y = \dfrac{1}{x - 2} + 3
Consider the graph of y = \dfrac{1}{x}:
How do we shift the graph of y = \dfrac{1}{x} to get the graph of y = \dfrac{1}{x} + 3 ?
How do we shift the graph of y = \dfrac{1}{x} to get the graph of y = \dfrac{1}{x + 2} ?
Sketch the graph of y=\dfrac{1}{x} + 3.
Sketch the graph of y=\dfrac{1}{x+2}.
Consider the function f \left( x \right) = \dfrac{3}{x}.
How can the graph of f \left( x \right) be obtained from the graph of y = \dfrac{1}{x} ?
Sketch the graph of f \left( x \right).
What is the domain of f \left( x \right)?
What is the range of f \left( x \right)?
Is the function f \left( x \right) increasing or decreasing over its domain?
Consider the function f \left( x \right) = \dfrac{1}{x + 4}.
How can the graph of f \left( x \right) be obtained from the graph of y = \dfrac{1}{x} ?
Sketch the graph of f \left( x \right).
What is the domain of f \left( x \right)?
What is the range of f \left( x \right)?
Is the function f \left( x \right) increasing or decreasing over its domain?
Consider the function y = \dfrac{2}{x + 1}.
State the domain of the function.
State the equation of the vertical asymptote.
Rearrange y = \dfrac{2}{x + 1} to make x the subject.
Hence, state the range of the function.
Hence, state the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the function y = \dfrac{3}{x} + 2.
State the domain of the function.
State the equation of the vertical asymptote.
Rearrange the equation to express x in terms of y.
State the range of the function.
Hence, state the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the function y = - \dfrac{3}{x - 1}.
State the domain of the function.
State the equation of the vertical asymptote.
Rearrange the equation to express x in terms of y.
Hence, state the range of the function.
State the equation of the horizontal asymptote.
Sketch the graph of the function.
Consider the function y = - \dfrac{3}{5 \left(x + 2\right)}.
State the value of a that completes the domain of the function:
Domain: x\in \Reals, x \neq a
State the equation of the vertical asymptote.
As x approaches \infty, what value does y approach?
Hence, state the equation of the horizontal asymptote.
State the value of b that completes the range of the function:
Range: x\in \Reals; y \neq b
Sketch the graph of the function.
The time, t, taken by a typist to type up a document is inversely proportional to his typing speed, s. That is, the quicker the typing speed, the less time it will take. If it takes a typist 20 minutes to type a particular document, typing at a speed of 61 words per minute:
Find the constant of variation k.
How long (in minutes) will it take a typist with a typing speed of 30.5 words per minute to type up the document?
The rent, electricity, telephone bill and other expenses for a flat cost a total of \$490 per week. These expenses are shared equally between the tenants of the flat.
How much will each occupant pay if the flat is shared by two people?
Let the number of occupants be x, and the cost paid by each occupant be y. Write a formula that relates the two variables.
Use this equation to fill in the following table of values. Round your answers correct to two decimal places where necessary.
| x | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y |
From the graph, what type of relationship exists between x and y?
A graph of the hyperbola y = \dfrac{10}{x} is shown:
Given points C\left( - 4 , 0\right) and D\left(2, 0\right), find the length of interval AB.