Consider the table of values:
| x | 9 | 18 | 27 | 36 |
|---|---|---|---|---|
| y | -68 | -131 | -194 | -257 |
Is y increasing or decreasing?
For every 1 unit increase in the x value, by how much does y change?
Hence, find the algebraic rule linking x and y.
Write an equation for v in terms of u for the following table of values:
| u | 0 | ... | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|
| v | -2 | ... | 54 | 62 | 70 | 78 |
Find the equation of the line:
That passes through \left(2, 7\right) and is parallel to the y-axis.
That passes through \left(3, 5\right) and is parallel to the x-axis.
That has a gradient of - 8 and crosses the y-axis at - 9.
That has gradient of - 2 and passes through the point \left( - 6 , - \dfrac{4}{3} \right).
If the equation ax + by = 18 represents a line with an x-intercept of - 6 and a y-intercept of 2, find the value of:
For each of the following pairs of points:
Find the gradient of the line that passes through the points.
Find the equation of the line that passes through the points.
\left(4, - 6 \right) and \left(6, - 9 \right)
\left(0, - 3 \right) and \left(1, 4\right)
\left(6, 5\right) and \left( - 16 , 11\right)
Consider the graph of the line:
What is the gradient?
What is the value of the y-intercept?
Write the equation of the line in gradient-intercept form.
The table shows some points on the line x = 0.
| x | 0 | 0 | 0 | 0 |
|---|---|---|---|---|
| y | -5.5 | -2.5 | 3.5 | 6.5 |
Does the line x = 0 represent the y-axis or the x-axis?
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver over 5 minutes:
| \text{Number of minutes passed }(x) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in metres }(y) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
What is the increase in depth each minute?
Write an equation for the relationship between the number of minutes passed (x) and the depth (y) of the diver.
In the equation y = 1.4 x, what does 1.4 represent?
At what depth would the diver be after 6 minutes?
Find how long the diver takes to reach 12.6 metres beneath the surface.
After Mae starts running, her heartbeat increases at a constant rate.
Write down the missing value from the table:
| \text{Number of minutes passed }(x) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate }(y) | 49 | 55 | 61 | 67 | 73 | 79 |
What is the unit change in y for the above table?
Form an equation that describes the relationship between the number of minutes passed, x, and Mae’s heartbeat, y.
In the equation y = 3 x + 49, what does 3 represent?
The number of fish in a river is approximated over a five year period. The results are shown in the following table:
| \text{Time in years }(t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Number of fish }(F) | 4800 | 4600 | 4400 | 4200 | 4000 | 3800 |
Sketch the graph for this relationship.
Write down the gradient of the line.
What does the gradient represent in this context?
What is the value of F when the line crosses the vertical axis?
Write down an equation for the line, using the given values.
Hence, determine the number of fish remaining in the river after 13 years.
Find the number of years until 2000 fish remain in the river by substituting F = 2000 into the equation and solve it for t.
A carpenter charges a callout fee of \$150 plus \$45 per hour.
Write an equation to represent the total amount charged, y, by the carpenter as a function of the number of hours worked, x.
What is the gradient of the function?
What does this gradient represent?
What is the value of the y-intercept?
What does the y-intercept represent in this context?
Find the total amount charged by the carpenter for 6 hours of work.
The graph shows the temperature of a room after the heater has been turned on for x minutes.
What is the gradient of the function?
What is the y-intercept?
Write an equation to represent the temperature of the room, y, as a function of time, x.
What does the gradient represent?
What does the y-intercept represent?
Find the temperature of the room after the heater has been turned on for 40 minutes.
A racing car starts the race with 150 litres of fuel. From there, it uses fuel at a rate of 5 litres per minute.
Complete the table of values:
| \text{ Number of minutes passed }(x) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \text{Amount of fuel left in tank }(y) |
Write an algebraic relationship linking the number of minutes passed (x) and the amount of fuel left in the tank (y).
After how many minutes, x, will the car need to refuel (i.e. when there is no fuel left)?
Consider the pattern for blue boxes attached:
Complete the table.
| \text {Number of columns, } c | 1 | 2 | 3 | 5 | 10 | 20 |
|---|---|---|---|---|---|---|
| \text{Number of blue boxes, } b |
Write a formula that describes the relationship between the number of blue boxes, b, and the number of columns, c.
How many blue boxes will there be if this pattern were to continue for 38 columns?
If this pattern continued and we had 45 blue boxes. How many columns would we have?
Kerry currently pays \$50 a month for her internet service. She is planning to switch to a fibre optic cable service.
Write an equation for the total cost T of Kerry's current internet service over a period of n months.
For the fibre optic cable service, Kerry pays a one-off amount of \$1200 for the installation costs and then a monthly fee of \$25. Write an equation of the total cost T of Kerry's new internet service over n months.
Complete the table for the total cost of the current internet service, given by T = 50 n:
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T (\text{ dollars}) |
Complete the table of values for the total cost of the fibre optic cable service, given by T = 25 n + 1200
| n | 1 | 6 | 12 | 18 | 24 |
|---|---|---|---|---|---|
| T (\text{ dollars}) |
Graph the lines for the total cost of Kerry's current internet service and the total cost of her new internet service on a number plane.
Using your graph to determine how many months it will take for Kerry to break even on her new internet service.