The equation x y = 7 can be described as a function where y varies inversely with x and has a constant of proportionality, k=7.
Find the constant of proportionality, k, for the following:
y = 4 when x = 3
State whether the following equations represent an inverse relationship between x and y:
The equation y = \dfrac{6}{x} represents an inverse relationship between x and y.
Write an equivalent equation for y = \dfrac{6}{x} which does not contain a fraction.
Can x or y be equal to 0?
When x = 2, 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{6}{x} lie?
The equation y = - \dfrac{12}{x} represents an inverse relationship between x and y.
Write an equivalent equation for y = - \dfrac{12}{x} which does not contain a fraction.
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?
For each hyperbola, find the x-value that corresponds to the given y-value:
State whether the following graphs are hyperbolas:
State whether each of the following is a feature of the graph of y = \dfrac{3}{x}:
Intercepts
Aysmptotes
Limits
Symmetry
Using technology or otherwise, graph the following functions and state:
The number of x-intercepts
The number of y-intercepts
Using technology or otherwise, graph the following functions and state whether the graph is increasing or decreasing for:
x > 0
x < 0
Using technology or otherwise, graph the two inverse relations on the same number plane and answer the following questions:
Do the graphs ever intersect?
For which values of x is the first function less than the second?
For which values of x is the first function greater than the second?
y = \dfrac{7}{x} and y = - \dfrac{7}{x}
y = \dfrac{2}{x} and y = \dfrac{4}{x}
y = - \dfrac{9}{x} and y = - \dfrac{36}{x}
Using technology or otherwise, graph the inverse relationship y = - \dfrac{5}{x}. Hence write the coordinates of the point of rotational symmetry.
Consider the function y = \dfrac{4}{x}.
Complete the table of values:
| x | 1 | 2 | 3 | 4 | 5 |
| y |
Is y = \dfrac{4}{x} increasing or decreasing when x > 0?
Describe the rate of change of the function when x > 0.
The equation y = - \dfrac{6}{x} represents an inverse relationship between x and y.
Complete the table of values:
| x | -5 | -4 | -3 | -2 | -1 |
| y |
Is y = - \dfrac{6}{x} increasing or decreasing when x < 0?
Describe the rate of change of the function when x < 0.
Consider the hyperbola that has been graphed.
Complete the statement: Every point \left(x, y\right) on the hyperbola is such that x y=⬚.
Determine whether the following statements are true or false:
As x increases, y increases.
As x decreases, y increases.
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.
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.
State whether the following relationships can be modelled by an inverse 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 people's height and weight.
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.
Sketch a graph of the relationship P = \dfrac {6000}{V} for 0 \leq V \leq 2000.
Find 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.
Sketch a graph of the relationship t = \dfrac{100}{S}.
Find the time taken if the speed travelled is 10 \text{ km/h}.
Find 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?
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.
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.