2.04 Hyperbolas and inverse variation
If y varies inversely with x, write an equation that uses k as the constant of variation.
If \left(10, 15\right) is on the curve P = \dfrac{k}{Q}, solve for k.
Determine whether each of the following is an example of a direct variation or an inverse variation:
The variation relating the distance between two locations on a map and the actual distance between the two locations.
The variation relating the number of workers hired to build a house and the time required to build the house.
If r varies inversely with a, write an equation that uses k as the constant of variation.
Determine whether the following equations represent an inverse relationship between x and y:
x = 1 + y^{3}
x = \dfrac {8}{y^{2}}
y = 6 x + 8
x y = - 7
x = \dfrac {2}{y}
x y = 5 x
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.
State whether the following tables 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 |
m is proportional to \dfrac {1}{p}. Consider the values in the table which represents the relationship.
| p | 4 | 6 | 7 | x |
|---|---|---|---|---|
| m | 63 | y | 36 | 28 |
Determine the constant of proportionality, k.
Find the values of x and y.
Find the equation relating t and s for each of 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} |
Use technology to graph the inverse relationship y = \dfrac {6}{x}.
How many x-intercepts does the graph have?
How many y-intercepts does the graph have?
Consider the graph of y = \dfrac {2}{x}.
For x > 0, as x increases, what does y approach?
For x > 0, as x approaches 0, what does y approach?
State the equation of the vertical asymptote.
State the equation of the horizontal asymptote.
The graph has two axes of symmetry. State their equations.
The equation y = - \dfrac {12}{x} represents an inverse relationship between x and y.
When x = 3, what is the value of y?
If x is a positive value, is the corresponding y value positive or negative?
If x is a negative value, is the corresponding y value positive or negative?
In which quadrants does the graph of y = \dfrac {- 12}{x} lie?
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}.
Complete the following statement:
As x approaches ⬚ from the right, the function value approaches \infty. As x approaches ⬚ from the left, the function value approaches -\infty.
As x approaches \infty and -\infty, y approaches ⬚. 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 does y approach?
As x approaches -\infty, what does y approach?
Consider the hyperbola shown:
Complete the statement:
"Every point \left(x, y\right) on the hyperbola is such that x y = ⬚."
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.
Consider the function graphed below:
What is the equation of the vertical asymptote?
What is the equation of the horizontal asymptote?
State the domain of the function.
State the range of the function.
For functions of the form f \left( x \right) = \dfrac {k}{x}, where k is a constant, the domain is always \left(-\infty, 0\right) \cup \left(0, \infty\right). What does this mean for the function when x = 0?
Consider the function y = \dfrac {1}{x}.
For what value of x is the function undefined?
What is the domain of y = \dfrac {1}{x}?
For what value of y is the function underfined?
What is the range of y = \dfrac {1}{x}?
Patricia determined the range of the function y = \dfrac {2}{x} is \left(-\infty, 0\right) \cup \left(0, \infty\right).
By filling in the gaps below, complete her reasoning.
Looking at y = \dfrac {2}{x}, notice that the numerator ⬚ is non-zero, and so \dfrac {2}{x} can never be equal to ⬚.
Another approach is to rearrange the equation and make x the subject. Then we get x = ⬚ . In this form we can see that the denominator ⬚ cannot be equal to ⬚.
Consider the functions y = \dfrac {4}{x} and y = \dfrac {2}{x}.
When x = 2, what is the value of y if y = \dfrac {4}{x}?
When x = 2, what is the value of y if y = \dfrac {2}{x}?
Which graph lies further away from the axes?
For hyperbolas of the form y = \dfrac {k}{x}, as the k value increases, what happens to the graph?
Consider the inverse variation equation 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 on a cartesian plane.
Ursula wants to sketch the graph of y = \dfrac {7}{x}, but knows that it will look similar to many other hyperbolas.
What can she do to the graph to show that it is the hyperbola y = \dfrac {7}{x}, rather than any other hyperbola of the form y = \dfrac {k}{x}?
For each of the following functions:
Complete a table of values of the form:
| x | -2 | -1 | -\dfrac{1}{2} | \dfrac{1}{2} | 1 | 2 |
|---|---|---|---|---|---|---|
| y |
Sketch the graph of the function.
State which quadrants the function lies in.
The relationship between the current, C, (in amperes) and resistance, R, (in ohms) in an electrical circuit is given by: C = \dfrac {240}{R}where the voltage provided to the circuit is 240 V.
Graph this function for 0 \leq R \leq 50.
What happens to the current as the resistance increases?
Consider the hyperbola that has been graphed. Points A \left(4, 2\right), B, C \left( - 4 , k\right) and D form the vertices of a rectangle.
Find the value of k.
Hence find the area of the rectangle with vertices ABCD.
Boyle's law describes the relationship between pressure and volume of a gas of fixed mass under constant temperature. The pressure for a particular gas can be found using P = \dfrac {6000}{V} where P has units \text{kg/cm}^2 and V has units \text{cm}^3.
Graph the relationship P = \dfrac {6000}{V} for 0 \leq V \leq 2000.
What is the pressure if the volume is 1\text{ cm}^3?
What happens to the pressure as the volume increases?
The time it takes a commuter to travel 100\text{ km} depends on how fast they are going. We can write this using the equation t = \dfrac{100}{S} where S is the speed in \text{ km/h} and t is the time taken in hours.
Graph the relationship t = \dfrac {100}{S}.
What is the time taken if the speed travelled is 10\text{ km/h}?
What is the time taken if the speed travelled is 50\text{ km/h}?
If we want the travel time to decrease, what must happen to the speed of travel?