5.10 Hyperbolas
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{3}{x}.
Complete the following table of values:
| x | -2 | -1 | -\dfrac{1}{2} | \dfrac{1}{2} | 1 | 2 |
|---|---|---|---|---|---|---|
| y |
Determine whether the following is a feature of the graph of y = \dfrac{3}{x}:
Intercepts
Aysmptotes
Limits
Symmetry
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{1}{2 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 following graph of y = \dfrac{4}{x}:
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?
\\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?
The graph has two axes of symmetry. State the equation of the lines of symmetry.
Consider the equation f \left( x \right) = \dfrac{3}{x}.
Sketch a graph of the function on a number plane.
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 equation y = \dfrac{8}{x} represents an inverse relationship between x and y.
Determine whether the equations below are equivalent to y = \dfrac{8}{x}:
x = 8 y
x y = 8
y = 8 x
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{8}{x} lie?
The equation y = - \dfrac{3}{x} represents an inverse relationship between x and y.
Determine whether the equations below are equivalent to y = -\dfrac{3}{x}:
x y = 3
y = - 3 x
x y = - 3
x = - 3 y
When x = 1, 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{3}{x} lie?
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?
What happens to the value of the fraction as x approaches -\infty?
Determine whether the following graphs demonstrates this behaviour for x > 0.
Consider the function y = \dfrac{6}{x}.
Complete the table of values:
| x | -6 | -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{6}{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 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.
Consider the functions y = \dfrac{4}{x} and y = \dfrac{3}{x}.
For y = \dfrac{4}{x}, when x = 1, what is the y-value?
For y = \dfrac{3}{x}, when x = 1, 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 graph of the hyperbola y = \dfrac{2}{x}:
What would be the new equation if the graph was shifted upwards by 2 units?
What would be the new equation if the graph was shifted to the right by 9 units?
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} + 4 ?
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} + 4.
Sketch the graph of y=\dfrac{1}{x+2}.
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 following hyperbolas:
y = \dfrac{6}{x} \text{ and } y = \dfrac{6}{x} - 5What is the y value of y = \dfrac{6}{x} corresponding to x = 2 ?
What is the y value of y = \dfrac{6}{x} - 5 corresponding to x = 2 ?
How is y = \dfrac{6}{x} transformed to make y = \dfrac{6}{x} - 5 ?
Consider the following hyperbolas:
y = \dfrac{4}{x} \text{ and } y = \dfrac{4}{x+3}What value cannot be substituted for x in y = \dfrac{4}{x} ?
In which quadrants does y = \dfrac{4}{x} lie?
What value cannot be substituted for x in y = \dfrac{4}{x+3} ?
In which quadrants does y = \dfrac{4}{x+3} lie?
How can the graph of y = \dfrac{4}{x} be transformed to create the graph of y = \dfrac{4}{x+3} ?
Consider the graphs of f \left(x\right) = \dfrac{3}{x} and g \left(x\right):
Write the equation to describe the transformation of g \left(x\right).
State the equation of g \left(x\right).
The following graph shows the relationship between the current (in amperes) and resistance (in ohms) in an electrical circuit, when the voltage provided to the circuit is 240\text{ V}.
The equation could be described as
\dfrac{240}{\text{Resistance}} = \text {Current}
What happens to the current as the resistance increases?
The time it takes a commuter to travel 100\text{ km} depends on how fast they are going. It can be written using the equation t = \dfrac{100}{S} where S is the speed in km/h and t is the time taken in hours.
Sketch the graph of 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?
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{6600}{V}, where P has units \text{ kg/cm}^2 and V has units \text{ cm}^3.
Sketch the graph of the relationship P = \dfrac{6600}{V}.
What is the pressure if the volume is 4\text{ cm}^3?
What happens to the pressure as the volume increases?