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 |
Sketch the graph on a number plane.
In which quadrants does the graph lie?
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 on a number plane.
In which quadrants does the graph lie?
Consider the function y = \dfrac{2}{x}
Complete the following table of values:
| x | -2 | -1 | \dfrac{- 1}{2} | \dfrac{1}{2} | 1 | 2 |
|---|---|---|---|---|---|---|
| y |
Sketch the graph on a number plane.
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 |
Sketch the graph on a number plane.
In which quadrants does the graph lie?
Consider the hyperbolic function y = \dfrac{3}{x} - 3.
Which of the following indicates 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?
Sketch the graph of y = \dfrac{3}{x} - 3
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}?
Consider the function y = \log_{4} x and its given graph:
Complete the following table of values:
| x | \dfrac{1}{16} | \dfrac{1}{4} | 4 | 16 | 256 |
|---|---|---|---|---|---|
| y |
Find the x-intercept.
How many y-intercepts does the function have?
Find the x-value for which \log_{4} x = 1.
Consider the function y = \log_{2} x.
Complete the following table of values:
| x | \dfrac{1}{2} | 1 | 2 | 4 | 16 |
|---|---|---|---|---|---|
| y |
Sketch a graph of the function.
State the equation of the vertical asymptote.
Consider the function y = \log_{4} x.
Complete the table of values.
| x | \dfrac{1}{1024} | \dfrac{1}{4} | 1 | 4 | 16 | 256 |
|---|---|---|---|---|---|---|
| y |
Is \log_{4} x an increasing or decreasing function?
Describe the behaviour of \log_{4} x as x approaches 0.
State the value of y when x = 0.
Consider the given graph of f \left( x \right) = \log_{k} x:
Determine the value of the base k.
Hence, state the equation of f \left( x \right).
Describe the following transformations:
The transformation of g \left( x \right) = \log_{10} x into f \left( x \right) = a \log_{10} x, where a > 1.
The transformation of g \left( x \right) = a \log_{10} x into f \left( x \right) = - a \log_{10} x.
This graph shows the relationship between the current, C, and the resistance, R, in an electrical circuit, when the voltage provided to the circuit is 240 V.
The equation of the function is C = \dfrac{240}{R}
Describe what happens to the current as the resistance increases?
The time it takes a commuter to travel 100 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 km/h and t is the time taken in hours.
Graph the relationship t = \dfrac{100}{S}.
Find the time taken if the speed travelled is 10 km/h.
Find the time taken if the speed travelled is 50 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, P \text{ kg/cm}^2 , and volume, V cm, of a gas of fixed mass under constant temperature. The pressure for a particular gas is given by P = \dfrac{6000}{V}.
Graph the relationship P = \dfrac{6000}{V}.
Find the pressure if the volume is 1\text{ cm}^3.
What happens to the pressure as the volume increases?