Consider the equation y = 7 x.
What is the value of y when x = - 5?
What is the value of y when x = 0?
What is the value of y when x = 5?
What is the value of y when x = 10?
Hence, complete the table of values:
| x | - 5 | 0 | 5 | 10 |
|---|---|---|---|---|
| y |
For each of the following, use the equation to complete the table of values:
y = x
| x | - 1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = 5 x
| x | - 10 | - 5 | 0 | 5 |
|---|---|---|---|---|
| y |
y = 5 x + 6
| x | - 10 | - 5 | 0 | 5 |
|---|---|---|---|---|
| y |
y = \dfrac{x}{8}
| x | - 4 | - 2 | 0 | 2 |
|---|---|---|---|---|
| y |
y = - 3 x + 8
| x | - 6 | - 3 | 0 | 3 |
|---|---|---|---|---|
| y |
y = 3 x - 4
| x | - 3 | -1 | 2 | 6 |
|---|---|---|---|---|
| y |
y = - 5 x
| x | - 4 | - 2 | -1 | 2 |
|---|---|---|---|---|
| y |
y = - 4 x - 3
| x | - 2 | -1 | 1 | 4 |
|---|---|---|---|---|
| y |
y = - \dfrac{x}{8}
| x | - 1 | 1 | 4 | 5 |
|---|---|---|---|---|
| y |
y = 4 x - 3
| x | - 4 | 0 | 4 | 8 |
|---|---|---|---|---|
| y |
For each of the following equations:
Construct a table of values for x=-1,0,1,2.
Hence, sketch the graph of the linear equation.
Consider the equation y = \dfrac{x}{2}.
Complete the table of values below:
| x | -6 | -4 | -2 | 0 |
|---|---|---|---|---|
| y |
Sketch the graph of y = \dfrac{x}{2}.
Consider the equation y = - \dfrac{x}{5}.
Complete the table of values below:
| x | -10 | -5 | 0 | 5 |
|---|---|---|---|---|
| y |
Sketch the graph of y = - \dfrac{x}{5}.
Consider the equation y = \dfrac{x}{2} + 5.
Complete the table of values below:
| x | -4 | -2 | 0 | 2 |
|---|---|---|---|---|
| y |
Sketch the graph of y = \dfrac{x}{2} + 5.
Consider the equation y = - \dfrac{x}{4} + 5.
Complete the table of values below:
| x | -8 | -4 | 0 | 4 |
|---|---|---|---|---|
| y |
Sketch the graph of y = - \dfrac{x}{4} + 5.
Consider the equation y = \dfrac{x}{6} - 5.
Complete the table of values below:
| x | -5 | -3 | 0 | 6 |
|---|---|---|---|---|
| y |
Sketch the graph of y = \dfrac{x}{6} - 5.
Consider the equation y = - \dfrac{x}{3} - 6.
Complete the table of values below:
| x | -3 | -2 | -1 | 0 |
|---|---|---|---|---|
| y |
Sketch the graph of y = - \dfrac{x}{3} - 6.
Complete the table of values for the following graphs:
| x | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| y |
| x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| y |
Consider the following graph:
Complete the table of values:
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y |
Does the point \left(3, 8\right) lie on the line?
Consider the following graph:
Complete the table of values:
| x | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| y |
Does the point \left(3, 5\right) lie on the line?
Consider the equation y = - x.
Complete the table of values:
Sketch a graph of the line.
Does the point \left( 2.5 , - 2.5 \right) lie on the line?
| x | - 1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
Consider the equation y = - x + 1.
Complete the table of values:
Sketch a graph of the line.
Does the point \left( 1.5 , - 0.5 \right) lie on the line?
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
Consider the following graph where the points form a linear pattern:
State the coordinates of the next point in the pattern to the right.
Complete the table of values for the pattern of points:
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y |
Write the rule for the line that would pass through the given points in the form y=x+c.
As x increases, does the value of y increase or decrease?
Consider the following graph, where the points form a linear pattern:
State the coordinates of the next point in the pattern to the right.
Complete the table of values for the pattern of points:
| x | 10 | 11 | 12 | 13 |
|---|---|---|---|---|
| y | 6 |
Write the rule for the line that would pass through the given points in the form y=x-c.
As x increases, does the value of y increase or decrease?
Consider the following equations:
Construct a table of values for x=-1,0,1.
Sketch the graph of the equation.
Consider the equation y = - x - 2.
Complete the table of values:
Sketch the graph of the line.
State the coordinates of the axes intercepts.
Find x when y = - 4.
| x | - 1 | 0 | 1 | 3 |
|---|---|---|---|---|
| y |
Amy is building a brick shed and starts on one of the side walls. Her progress is shown:
Complete the table of values:
| Hours | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Number of layers |
The height of a candle is measured at 15-minute intervals.
Complete the table of values with the height of the candle:
| Time (minutes) | 15 | 30 | 45 | 60 |
|---|---|---|---|---|
| Height (cm) |
Complete the table of values for the figures in the pattern:
| Step number | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| Number of squares |
Complete the table of values for the matchstick figures in the pattern:
| Step number | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| Number of matches |
After each week, the number of apples on an apple tree increase according to the given pictures.
Complete the table of values:
| Week | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Number of apples |
The following figures were made from matchsticks of equal length:
Complete the table of values for the figures:
| Step number | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Number of matchsticks |
A racing car starts the race with 140 litres of fuel. From there, it uses fuel at a rate of 2 litres per minute. Complete the table of values:
| \text{Number of minutes passed} \,(x) | 0 | 5 | 10 | 15 | 20 | 70 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left}\, (y) |
There are 20 \text{ L} of water in a rainwater tank. It rains for a period of 24 hours and during this time the tank fills up at a rate of 8 \text{ L/h}. Complete the table of values:
| \text{Number of hours passed }(x) | 0 | 4 | 6 | 7 | 9 | 11 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank }(y) |
The cost of a taxi ride C is given by C = 2.50 t + 3 where t is the duration of the trip in minutes.
Complete the table of values:
| \text{Time in minutes }(t) | 6 | 7 | 8 | 9 | 11 | 16 |
|---|---|---|---|---|---|---|
| \text{Cost in dollars }(C) |
Buzz recorded his savings (in dollars) over a few months in the graph given.
Complete the table:
| \text{Months} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Savings } \left(\$\right) |
Is Buzz correct if he estimates that he will have exactly \$60 in his savings by month 5?
The graph shows the relationship between the number of cartons and the total number of eggs in them.
Complete the table:
| \text{Cartons} | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Eggs} |
A dam used to supply water to the neighboring town had the following data recorded for its volume over a number of months:
| \text{Month }(M) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Volume in billions of litres } (V) | 112 | 106 | 110 | 80 |
Is this relationship linear?
Explain a method to check whether the relationship is linear, without having to plot the points.
A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table below shows the depth of the diver over 4 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 |
Find the increase in depth each minute.
Write a linear equation of the form y=mx for the relationship between the number of minutes passed, x, and the depth, y, of the diver.
Calculate the depth of the diver after x= 6 minutes.
Find the number of minutes, x, it takes the diver to reach y=12.6 metres beneath the surface.
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:
| \text{Number of litres }(x) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol }(y) | 0 | 16.40 | 32.80 | 49.20 | 65.60 |
Find the cost of petrol per litre.
Write an equation of the form y=mx for the relationship between the number of litres of petrol pumped \left(x\right) and the cost of the petrol \left(y\right).
Calculate the cost, y, of x=47 \text{ L} of petrol.
How many litres of petrol, x, can be bought for y=\$32.80?
Consider the following table that shows the temperature of a metal plate, in \degree\text{C}, after an amount of time, measured in minutes:
| \text{Time }(x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Temperature }(y) | 10 | 15 | 20 | 25 | 30 |
Graph the linear relationship represented in the table.
By how much is the temperature increasing each minute?
Find the initial temperature.
Hence, form an equation relating x and y.
Find the temperature of the plate after 12 minutes.
The number of calories burned by the average person while dancing is modelled by the equation C = 8 m, where m is the number of minutes.
Sketch the graph of this equation to show the calories burnt after each 15-minute interval.
After Mae starts running, her heart rate in beats per minute increases at a constant rate as shown in the following table:
| \text{Number of minutes passed, }x | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate, }y | 49 | 55 | 61 | 67 | 73 | 79 |
Determine Mae's heart rate after 12 minutes.
Calculate the change in heart rate per minute.
Write an equation that describes the relationship between the number of minutes passed, x, and Mae’s heart rate, y.
Explain the meaning of the y-intercept in this context.
Consider the pattern for blue boxes below:
Complete the table:
| \text{Number of columns } (x) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \text{Number of blue boxes } (y) | 1 | 3 |
Write a formula that describes the relationship between the number of blue boxes (y) and the number of columns (x) in the form y=mx-c.
Find the number of blue boxes, y, required for:
38 columns
92 columns
Find the number of columns, x, that would contain:
45 blue boxes
51 blue boxes
A racing car starts the race with 250 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 | 50 |
|---|---|---|---|---|---|---|
| \text{Amount of fuel left in tank, }y |
Determine an algebraic rule linking the number of minutes passed, x, and the amount of fuel left in the tank, y.
Explain the meaning of the gradient in this context.
The table shows the linear relationship between the number of plastic chairs manufactured, x, and the total manufacturing cost, y:
| Number of plastic chairs | 5 | 10 | 15 |
|---|---|---|---|
| Cost (dollars) | 135 | 185 | 235 |
State the gradient of the linear function.
Form an equation relating x and y.
Find the y-intercept.
Find the total cost of manufacturing 25 plastic chairs.
Explain the meaning of the y-intercept in this context.
Explain the meaning of the gradient of the function in this context.
A ball is rolled down a slope. The table below shows the velocity (V) of the ball after a given number of seconds (t):
| \text{Time in seconds } (t) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Velocity in m/s } (V) | 12 | 13.3 | 14.6 | 15.9 | 17.2 | 18.5 |
Plot the graph of the ball's velocity against time on a coordinate plane.
Calculate the gradient of the line.
What does the gradient represent in this context?
State the vertical axis intercept of the line.
What does the vertical axis intercept represent in this context?
Write an algebraic equation for the line, expressing V in terms of t.
Hence, determine the velocity of the ball after 19 seconds. Round your answer to one decimal place.
A baseball is thrown vertically upward by a baseball player when he is standing on the ground, and the velocity of the baseball V (in metres per second) after T seconds is given by V = 120 - 32 T.
Complete the table of values:
| \text{Time} | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Vertical Velocity} |
State the gradient of the linear function.
Explain the negative value of V when T = 4.
It starts raining and an empty rainwater tank fills up at a constant rate of 2 litres per hour. By midnight, there are 20 litres of water in the rainwater tank. As it rains, the tank continues to fill up at this rate.
Complete the table of values:
| \text{Number of hours passed since midnight } (x) | 0 | 1 | 2 | 3 | 4 | 4.5 | 10 |
|---|---|---|---|---|---|---|---|
| \text{Amount of water in tank } (y) |
Plot the graph depicting the situation on a coordinate plane.
Write an algebraic relationship linking the number of hours passed since midnight (x) and the amount of water in the tank (y).
Determine the y-intercept of the line.
At what time prior to midnight was the tank empty?
In a study, scientists found that the more someone sleeps, the quicker their reaction time. The table below displays the findings:
| \text{Number of hours of sleep } (x) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| \text{Reaction time in seconds } (y) | 6 | 5.8 | 5.6 | 5.4 | 5.2 | 5 |
How much does the reaction time decrease for each extra hour of sleep?
Write an algebraic equation relating the number of hours of sleep (x) and the reaction time (y).
Calculate the reaction time for someone who has slept 4.5 hours.
Calculate the number of hours someone sleeps if they have a reaction time of 5.5 seconds.
A racing car starts the race with 150 \text{ L} of fuel. From there, it uses fuel at a rate of 5\text{ L} per minute.
Complete the following table of values:
| \text{Number of minutes passed } (x) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \text{Amount of fuel left in the tank } (y) |
Write an algebraic relationship linking the number of minutes passed \left(x\right) and the amount of fuel left in the tank \left(y\right).
How many minutes will it take for the car to run out of fuel?
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 a graph that corresponds to this information.
Calculate the gradient of the line.
What does the gradient represent in this context?
State the value of F when the line crosses the vertical axis.
Determine 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, \left(t\right), until 2000 fish remain in the river.
A car travels at an average speed of 75\text{ km/h}.
Complete the table of values for D = 75 t, where D is the distance travelled in kilometres and t is the time taken in hours:
| t | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| D |
How far will the car travel in 9 hours?
Sketch the graph of D = 75 t on a coordinate plane.
State the gradient of the line.
If the destination is 675\text{ km} ahead, how long would it take for the car to reach the destination at the given speed?
The graph shows the relationship between water temperatures and surface air temperatures:
Complete the table of values:
| \text{Water Temperature } \left(\degree \text{C} \right) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \text{Surface Air Temperature } \left(\degree \text{C} \right) |
Write an algebraic equation representing the relationship between the water temperature (x) and the surface air temperature (y).
Find the surface air temperature when the water temperature is 14 \degree \text{C}.
Find the water temperature when the surface air temperature is 23 \degree \text{C}.