11.03 Hyperbolas (Extended)
If y varies inversely with x, write the equation for y in terms of x.
For each of the following relationships, write an equation involving constant k:
r is inversely proportional to c.
h is inversely proportional to the cube of m.
Determine whether each table below could represent an inversely proportional relationship between x and y:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 3 | 1.5 | 1 | 0.75 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 36 | 18 | 12 | 9 |
| x | 1 | 5 | 6 | 10 |
|---|---|---|---|---|
| y | 3 | 75 | 108 | 300 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 4 | 5 | 6 | 7 |
Consider the equation s = \dfrac{375}{t}.
State the constant of proportionality.
Find the exact value of s when t = 6.
Find the exact value of s when t = 12.
m is proportional to \dfrac{1}{p}. Consider the values in the table which represents this relationship:
Determine the constant of proportionality, k.
Find the values of x and y in the table.
| p | 4 | 6 | 7 | x |
|---|---|---|---|---|
| m | 63 | y | 36 | 28 |
Find the equation relating t and s for the following tables of values:
| s | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| t | 48 | 24 | 16 | 12 |
| s | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| t | \dfrac{2}{9} | \dfrac{1}{18} | \dfrac{2}{81} | \dfrac{1}{72} |
If a is inversely proportional to x, and a = 20 when x = 10:
Find the constant of variation, k.
Express a in terms of x.
Find the value of a when x = 5.
The equation y = - \dfrac{12}{x} represents an inverse relationship between x and y.
Determine whether the equations below are equivalent to y = - \dfrac{12}{x}:
y = - 12 x
x y = - 12
x = - 12 y
x y = 12
When x = 3, what is the value of y?
If x is a positive value, must the corresponding y value be positive or negative?
If x is a negative value, must the corresponding y value be positive or negative?
In which quadrants does the graph of y = \dfrac{- 12}{x} lie?
Find the equation of the curve that passes through the points in the table:
| x | -2 | -1 | 1 | 2 |
|---|---|---|---|---|
| y | \dfrac{1}{2} | 1 | -1 | -\dfrac{1}{2} |
Consider the functions y = \dfrac{4}{x} and y = \dfrac{2}{x}.
For y = \dfrac{4}{x}, when x = 2, what is the y-value?
For y = \dfrac{2}{x}, when x = 2, what is the y-value?
Which graph lies further away from the axes?
For hyperbolas of the form y = \dfrac{k}{x}, what happens to the graph as the value of k increases?
Consider the function y = \dfrac{1}{x} which is defined for all real values of x except 0.
Complete the following table of values:
| x | -2 | -1 | -\dfrac{1}{2} | -\dfrac{1}{4} | \dfrac{1}{4} | \dfrac{1}{2} | 1 | 2 |
|---|---|---|---|---|---|---|---|---|
| y |
Plot the points from the table of values.
Hence, sketch the curve on the same axes.
In which quadrants does the graph lie?
Consider the function y = - \dfrac{1}{x}.
Complete the following table of values:
| x | -2 | -1 | -\dfrac{1}{2} | \dfrac{1}{2} | 1 | 2 |
|---|---|---|---|---|---|---|
| y |
Plot the points from the table of values.
Hence, sketch the curve on the same axes.
In which quadrants does the graph lie?
Consider the function y = \dfrac{6}{x}.
Complete the following tables of values:
| x | \dfrac{1}{4} | \dfrac{1}{2} | 1 | 2 | 4 |
|---|---|---|---|---|---|
| y |
Plot the points from the table of values.
Consider the expression \dfrac{2}{x} for x > 0.
What happens to the value of the fraction as x increases?
What happens to the value of the fraction as x approaches 0?
Determine whether the following graphs demonstrates this behaviour for x > 0.
Consider the graph of y = \dfrac{2}{x} below:
For positive values of x, as x increases, what value does y approach?
As x takes small positive values approaching 0, what value does y approach?
What are the values that x and y cannot take?
The graph has two axes of symmetry. State the equation of the lines of symmetry.
Consider the graph of the function y = \dfrac{4}{x}, \\x = 0 and y = 0 are lines that the curve approaches very closely as x gets very small and very large.
What is the name of such lines?
Consider the function y = \dfrac{2}{x}.
Complete the table of values:
| x | -2 | -1 | -\dfrac{1}{2} | -\dfrac{1}{10} | -\dfrac{1}{100} | \dfrac{1}{100} | \dfrac{1}{10} | \dfrac{1}{2} | 1 |
|---|---|---|---|---|---|---|---|---|---|
| y |
For what value of x is the function undefined?
Rewrite the equation to make x the subject.
For what value of y is the function undefined?
Below is the graph of y = \dfrac{2}{x}.
What value should x approach from the right for the function value to approach \infty?
What value does the function approach as x approaches 0 from the left?
What value does y approach as x approaches \infty and -\infty? This is called the limiting value of the function.
Consider the function y = - \dfrac{5}{x}.
For what value of x is the function undefined?
As x approaches 0 from the positive side, what does y approach?
As x approaches 0 from the negative side, what does y approach?
As x approaches \infty, what value does y approach?
As x approaches -\infty, what value does y approach?
Determine whether the following is a feature of the graph of y = \dfrac{3}{x}:
Intercepts
Aysmptotes
Limits
Symmetry
Consider the hyperbola graphed below:
Every point \left(x, y\right) on the hyperbola is such that xy = a. What is the value of a?
When x increases, does y increase or decrease?
Determine whether the following relationships can be modelled by a function of the form x y = a.
The relationship between the number of people working on a job and how long it will take to complete the job.
The relationship between the number of sales and the amount of revenue.
The relationship between height and weight.
Ursula wants to sketch the graph of y = \dfrac{7}{x}, but knows that it will look similar to many other hyperbolas. How can she show that it is the hyperbola y = \dfrac{7}{x}, rather than any other hyperbola of the form y = \dfrac{k}{x}?
Consider the equation f \left( x \right) = \dfrac{4}{x}.
Sketch a graph of the function.
What type of symmetry does the graph have?
Find an expression for f \left( - x \right).
Does this verify that the function is rotationally symmetric about the origin?
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.
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.
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.