2.01 Graphs of linear relationships
Do the points on each of the following number planes form a linear relationship? Explain your answer.
Consider the three points A \left(3, 6\right), B \left(1, 0\right), and C \left(2, 3\right).
Plot the points on a number plane.
Do the points on each of the following number planes form a linear relationship? Explain your answer.
Consider the three points X \left(3, - 7 \right), Y \left(4, - 9 \right), and Z \left(1, - 2 \right).
Plot the points on the same number plane.
Do the points on each of the following number planes form a linear relationship? Explain your answer.
For each of the following equations and their table of values:
Plot the points from the table on a number plane.
Determine whether the points form a linear relationship.
y = x + 5
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 6 | 7 | 8 | 9 |
y = x
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 1 | 2 | 3 | 4 |
y = 3
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y | 3 | 3 | 3 | 3 |
y = x^{2}
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y | 4 | 1 | 0 | 1 |
y = x + 4
| x | -4 | -3 | -2 | -1 |
|---|---|---|---|---|
| y | 0 | 1 | 2 | 3 |
x = - 6
| x | -6 | -6 | -6 | -6 |
|---|---|---|---|---|
| y | -6 | -3 | 0 | 3 |
y = - x
| x | -3 | -2 | -1 | 0 | 1 |
|---|---|---|---|---|---|
| y | 3 | 2 | 1 | 0 | -1 |
y = 4x
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | -8 | -4 | 0 | 4 | 8 |
For each of the following equations:
Complete the table of values provided.
Plot the points from the table on a number plane.
y = 3 x
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
y = x - 5
| x | 4 | 5 | 6 | 7 |
|---|---|---|---|---|
| y |
y = x + 3
| x | 1 | 3 | 7 | 10 |
|---|---|---|---|---|
| y |
y = 2 x + 7
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y |
y = 3
| x | -6 | -3 | -2 | 2 |
|---|---|---|---|---|
| y |
y = - 2x + 1
| x | -2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y |
Consider the given table of values:
Plot the points in the table on a number plane.
Is the relationship between x and y linear?
| x | - 3 | - 2 | -1 | 0 |
|---|---|---|---|---|
| y | 15 | 10 | 5 | 0 |
If we extend the table of values to include x = 1, find the corresponding value of y.
Consider the given table of values:
Plot the points in the table on a number plane.
Is the relationship between x and y linear?
| x | - 8 | - 4 | 0 | 4 |
|---|---|---|---|---|
| y | - 2 | -1 | 0 | 1 |
If we extend the table of values to include x = 8, find the corresponding value of y.
Consider the given table of values:
Plot the points in the table on a number plane.
Is the relationship between x and y linear?
| x | - 1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | - 3 | 1 | 5 | 9 |
If we extend the table of values to include x = 3, find the corresponding value of y.
Consider the given table of values:
Does the ordered pair \left( - 4 , 8\right) satisfy the linear relationship between x and y?
| x | - 2 | -1 | 0 | 1 |
|---|---|---|---|---|
| y | 2 | -1 | - 4 | - 7 |
After each week, the number of apples on an apple tree increases according to the diagram:
Complete the following table of values:
| Week | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| No. Apples |
Plot the number of apples against each week on a number plane.
Is the relationship between the number of weeks and the number of apples linear?
The following pattern was created using matchsticks. Each figure is labelled with a number:
Complete the following table of values:
| Figure number | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| No. Matchsticks |
Plot the number of matchsticks used at each step on a number plane.
Is the relationship between the step number and the number of matchsticks linear?
A staircase with 5 steps is shown below. The height of each step is 15\text{ cm} and the length of each step is 10\text{ cm}.
Complete the following table of values:
| No. Steps | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| Height of staircase (cm) | 30 |
Plot the height of a staircase against the number of steps on a number plane.
Is the relationship between the height of a staircase and the number of steps linear?
Is it possible to have a staircase with height 105 \text{ cm} with steps that have the same height and length as in the image?
Complete the following table of values:
| No. Steps | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| Length of staircase (cm) | 20 |
Plot the length of a staircase against the number of steps on a number plane.
Is the relationship between the length of a staircase and the number of steps in a staircase linear?
Is it possible to have a staircase with length 70 cm with steps that have the same height and length as in the image?
A water tank initially starts with 65 L of water. Each day, 10 L of water is taken from the water tank.
Plot the volume (in L) of water in the tank against the number days, from 0 to 4 days.
Is the relationship between the volume of water in the water tank and the number of daysd linear?
How many litres of water will there be after 5 days?
The points on the number plane show the relationship between water temperature, x, and surface air temperature, y:
Complete the table of values:
| \text{Water Temperature in } \degree \text{C},\, x | - 3 | - 2 | - 1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| \text{Surface Air Temperature in } \degree \text{C},\, y |
Find the surface air temperature when the water temperature is 4 \degree \text{C}.
Find the surface air temperature when the water temperature is 14 \degree \text{C}.
Let the height of a candle be h \text{ cm}. If the candle is lit, the height of the candle decreases according to the equation \\ h = - 2 t + 9, where t is the time in minutes. A table of values is shown:
Plot the points in the table on a number plane.
| \text{Time} \\ \text{(}t \text{ mins)} | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Height of candle} \\ \text{(}h\text{ cm)} | 9 | 7 | 5 | 3 |
A dead tree is 7 \text{ m} tall and its height does not change over time. Let the height of the tree be y metres and the time passed be x years.
State whether the following coordinates satisfy the growth of the tree, where the x-coordinate is the time passed, and the y-coordinate is the height of the tree:
Plot the points from part (a) that satisfy the growth of the tree on a number plane.
Let the volume of water in a tank be V \text{ L}, and the time taken to fill up the tank be t minutes. Suppose the tank is initially empty and the number of litres in the tank is always three times the number of minutes that have passed.
Complete the table of values:
| \text{Time (}t \text{ mins)} | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Volume of water (}V \text{ L)} |
Which of the following equations represents the relationship between the time taken, t, and the volume of the tank, V?
V = 3
V = 3 t
V = 3 t + 2
V = t + 3
Plot the points in the table on a number plane.
Let the height of a pot plant be y \text{ cm}. As the pot plant grows, the height increases according to the equation y = 2 t + 9, where t is the elapsed time in weeks.
Complete the table of values:
| \text{Time (}t \text{ weeks)} | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \text{Height of pot plant (}y \text{ cm)} |
Plot the points in the table on a number plane.
Max is 21 years old and Sebastian is 14 years old. Let M represent Max's age and S represent Sebastian's age.
Write an equation for M in terms of S.
State whether following ordered pairs, in the form \left(S, M\right), satisfy the relationship between Max and Sebastian's ages:
Plot the points from part (b) that satisfy the relationship between their ages on a number plane.