Consider the values in each table. State whether they could represent a directly proportional relationship between x and y:
| x | 1 | 3 | 5 | 7 |
|---|---|---|---|---|
| y | 50 | 40 | 30 | 20 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 20 | 45 | 80 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 5 | 10 | 15 | 20 |
| x | 1 | 5 | 6 | 20 |
|---|---|---|---|---|
| y | 100 | 75 | 50 | 25 |
The diagram shows the graph of a straight line with positive gradient.
As x decreases towards -\infty, describe what happens to y?
As x increases towards \infty, describe what happens to y?
A straight line graph has a positive y-intercept and a positive x-intercept. Determine whether the following statements are true of this line graph.
Is the straight line graph increasing or decreasing?
Can we determine whether the straight line graph is steeper than y = x? Explain your answer.
For the following equations:
y = - 2 x
y = - 1 - \dfrac{9 x}{2}
For the following equations:
y = 6 \left( 3 x - 2\right)
5 x - 30 y - 25 = 0
Find the gradient of the line that passes through the given points:
\left( - 3 , - 1 \right) and \left( - 5 , 1\right)
\left( - 3 , 4\right) and \left(1, 4\right)
\left(2, - 6 \right) and the origin
Find the gradient of the line going through points A and B.
We want to determine if the points A \left(3, - 2 \right), B \left(5, 4\right) , and C \left(1, - 8 \right) are collinear.
Find the gradient of the line through A and B.
Find the gradient of the line through A and C.
Hence, state the gradient of the line through B and C.
Are the points A, B and C collinear?
Points P(-1,-1), Q(0, 1), R(-1, 6) andS(-2, 4) are plotted on the number plane shown. What type of quadrilateral is PQRS? Justify your answer with mathematical working.
Find the value of the unknown given the following:
A line passing through the points \left( - 1 , 4\right) and \left( - 4 , t\right) has a gradient equal to - 3.
A line passes through the points \left(11, c\right) and \left( - 20 , 16\right) and has a gradient of - \dfrac{4}{7}.
Explain why the gradient of a vertical line is undefined.
If the gradient of a line is zero, what does this tell us about the line?
Do the graphs of 2 x - 5 y = - 10 and - 2 x + 5 y = 10 have the same intercepts? Explain your answer.
Consider the graphed line below:
What is the gradient of the line?
Does this line have a y-intercept?
What is the x-intercept of the line?
What is the gradient of any line parallel to the x-axis?
Sketch the following lines on the same number plane:
y = - 6
x = - 1.5
y = 5
x = 2
State whether the following ordered pairs pass through the line y-5 = 3(x + 4):
Does \left(5, \dfrac{19}{3}\right) lie on the line y = \dfrac{2 x}{3} + 3 ?
Does \left(3, 10 \right) lie on the line y = 8x-15 ?
Does \left(-4, 36 \right) lie on the line y = -6x+12 ?
Find the value of the pronumeral given the following:
The point \left(5, q\right) satisfies the equation y = - 3 x - 5.
The point \left(a, a\right) satisfies the equation y = - 4 x - 5.
Determine whether the following is a graph of a linear relationship.
What makes the graph linear?
Consider the equation y = - x.
Complete the table of values for this equation:
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
Sketch the graph of the line..
Determine whether the point (2.5, -2.5) lies on the line.
Consider the equation y = - x + 1.
Complete the table of values for this equation.
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y |
Sketch the graph of the line..
Determine whether the point (1.5, -0.5) lies on the line.
For each of the following lines:
Find the x-intercept.
Find the y-intercept.
Sketch the graph.
Sketch the graphs of the following lines:
6 x + 2 y - 12 = 0
y = 4x - 1
A line with gradient, m = - 3 that passes through the point, \left( - 2 , 4\right)
A line with y-intercept =-1 and gradient m = \dfrac{5}{4}
Consider the following linear equations:
- y = 2 x + 2
- y = - 2 x + 2
Find the gradient and y-intercept of the line y = 2 x + 2.
Find the axes intercepts of the line y = -2 x + 2.
Sketch the graph of the lines of the 2 equations on the same number plane.
State the values of x and y which satisfy both equations.
Consider the following linear equations:
- y = - 4 x - 8
- y = - 2 x - 4
Find the axes intercepts of the line y = - 4 x - 8.
Find the axes intercepts of the line y = - 2 x - 4.
Sketch the graph of the lines of the 2 equations on the same number plane.
State the values of x and y which satisfy both equations.
Consider the following ramp:
What is the gradient of this skateboard ramp if it rises 0.9 metre above the ground and runs 1 metre horizontally at the base?
This ramp can only be used as a 'beginner’s ramp' if for every 1 metre horizontal run, it has a rise of at most 0.5 metre. Can it be used as a 'beginner’s ramp'?
A\left(2, 1\right), B\left(7, 3\right) and C\left(7, - 5 \right) are the vertices of a triangle.
Which side of the triangle is a vertical line?
Determine the area of the triangle using A = \dfrac{1}{2} b h.
The number of calories burned by the average person while dancing is modelled by the equation C = 8m, where m is the number of minutes.
Sketch the graph of this equation to show the calories burnt after each 15-minute interval.
The number of university students studying computer science in a particular country is modelled by the equation S = 4 t + 12, where t is the number of years since 2000 and S is the number of students in thousands.
Sketch the graph of this equation to show the number of computer science students each year from 2000 until 2020.
The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M = 1050 - 15 t.
After 61 hours, how many milligrams of medication are left in the body?
How many hours will it take for the medication to be completely removed from the body?
Avril is looking to find the best deal for an internet service to her home. Company A is offering a \$25 per month deal where there is no limit to her internet usage. Company B is offering rates according to the amount of internet usage (measured in GB), as shown in the table.
| Usage (GB) | Cost (dollars) |
|---|---|
| 5 | 17.50 |
| 10 | 20.00 |
| 20 | 25.00 |
| 50 | 40.00 |
Find the amount of internet usage in one month that would make the two offers cost the same amount.
Avril estimates that she will be using about 19 GB a month. Which company offers her the better deal?
David decides to start his own yoga class. The cost and revenue of running the class have been graphed:
How much revenue does David make for each student?
How many students must attend his class so that David can cover the costs?
How much profit does David make if there are 8 students in his class?
A\left(9, 3\right) is the point of intersection of the two lines:
-
y = - x + b
-
y = n x - b
Find the value of b.
Find the value of n.
A clothing manufacturer is deciding whether to employ people or to purchase machinery to manufacture their line of t-shirts. After conducting some research, they discover that the cost of employing people to make the clothing is y = 400 + 60 x, where y is the cost and x is the number of t-shirts to be made, while the cost of using machinery (which includes the cost of purchasing the machines) is y = 1000 + 30 x.
Graph the two lines on the same number plane.
State the value of x at which it will cost the same whether the t-shirts are made by people or by machines.
State the range of values of x for which it will be more cost efficient to use machines to manufacture the t-shirts.
State the range of values of x at which it will be more cost efficient to employ people to manufacture the t-shirts.
A gym offers aerobics classes where non-members pay \$3 per class and members pay a \$4 fee plus an additional \$2 per class. The monthly cost, y, of taking x classes can be modelled by the linear system of equations:
- Non-members: y = 3x
- Members: y = 2x + 4
Find the y-coordinates of the points on the line y = 3 x with the following x-coordinates:
What are the axes intercepts of the line y = 2 x + 4?
Graph the two lines on the same number plane.
State the values of x and y which satisfy both equations.
What do the coordinates of the solution mean in context?