11.01 Directly proportional functions
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 | 20 | 16 | 12 | 8 |
| x | 1 | 5 | 6 | 20 |
|---|---|---|---|---|
| y | 16 | 12 | 8 | 4 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 2 | 8 | 18 | 32 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 2 | 4 | 6 | 8 |
State whether the following graphs indicate that y is directly proportional to x:
Consider the equation y = 6 x.
State the gradient.
Sketch the graph of the equation.
Consider the equation y = -4x+9.
State the gradient.
Sketch the graph of the equation.
a is directly proportional to b and a = 54 when b = 9.
Graph the relationship between a and b.
Find the gradient of the line.
Express a in terms of b.
Find the value of a when b = 7.
Find the value of b when a = 60.
q is directly proportional to r and q = 10 when r = 20.
Graph the relationship between q and r.
Find the gradient of the line.
Express q in terms of r.
Find the value of q when r = 30.
Find the value of r when q = 22.
Given that y = 2 x, find the missing values in the table:
| x | 3 | |
|---|---|---|
| y | 14 |
Given that y = -4x + 3, find the missing values in the table:
| x | -3 | |
|---|---|---|
| y | 19 |
Find the equation relating to the given variables shown in the tables:
| a | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| b | 0 | 9 | 18 | 27 |
| n | 0 | 3 | 6 | 9 |
|---|---|---|---|---|
| r | 0 | \dfrac{9}{4} | \dfrac{9}{2} | \dfrac{27}{4} |
| x | 0 | 4 | 9 | 16 |
|---|---|---|---|---|
| y | -10 | -2 | 8 | 22 |
| s | 0 | 5 | 10 | 15 |
|---|---|---|---|---|
| t | 0 | \dfrac{1}{3} | \dfrac{2}{3} | 1 |
Consider the equation P = 70 t.
State the constant of proportionality.
Find the value of P when t = 4.
Consider the equation M = \dfrac{1}{10} n.
State the constant of proportionality.
Find the value of M when n = 50.
Consider the proportional relationships shown in the tables:
Sketch the graph of the relationship.
Find the unit rate of the relationship.
| x | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| y | 9.5 | 19 | 28.5 | 38 | 47.5 |
| x | 10 | 20 | 30 | 40 | 50 |
|---|---|---|---|---|---|
| y | 7.5 | 15 | 22.5 | 30 | 37.5 |
| x | 15 | 25 | 35 | 45 | 55 |
|---|---|---|---|---|---|
| y | 36 | 60 | 84 | 108 | 132 |
| x | 1 | 3 | 5 | 7 | 9 |
|---|---|---|---|---|---|
| y | 1.25 | 3.75 | 6.25 | 8.75 | 11.25 |
If y varies directly with x, and y = \dfrac{4}{5} when x = 3:
Find the constant of proportionality, k.
Find the equation of variation of y in terms of x.
If s varies directly with t, and s = \dfrac{3}{2} when s = 6:
Find the constant of proportionality, k.
Find the equation of variation of t in terms of s.
Write an equation involving constant k such that m varies directly with p.
Write an equation involving constant k for the revenue of a company, r, that varies directly with the number of sales, n.
Derek and Yuri are building robot kits. Derek can make 2 robots in 3 hours. Yuri can make 3 robots in 6 hours.
Sketch the graph of this proportional relationship. Assume both Derek and Yuri build robots at a constant rate.
Who is the faster builder?
Paul paints 15 plates every 6 hours.
Complete this proportion table:
| Plates painted | 0 | 15 | 30 | 60 | |
|---|---|---|---|---|---|
| Hours worked | 6 | 12 | 18 |
Hence, sketch the graph of this proportional relationship.
William is making a fruit smoothie. The amount of bananas and strawberries he uses is shown in the following proportion table:
| Strawberries | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| Bananas | 3.5 | 7 | 10.5 | 14 | 17.5 |
Sketch the graph of the proportional relationship.
Find the unit rate of the relationship.
Write two statements that describe the proportional relationship.
Harry and Carl love reading. Harry reads 16 books every 12 weeks. Carl has kept a table of his reading habits which is shown below:
| Number of weeks | 12 | 24 | 36 | 48 |
|---|---|---|---|---|
| Number of books read | 20 | 40 | 60 | 80 |
Complete the following table for Harry:
| Number of weeks | 12 | 48 | 60 | ||
|---|---|---|---|---|---|
| Number of books read | 16 | 32 | 48 | 80 |
Determine who reads more quickly.
Jimmy is saving money for a vacation. He knows that he can represent his savings over time using the equation y = \dfrac{11}{2} x, where x represents the number of days and y represents the savings. Ben saves \$7 every 4 days.
Complete the table showing Ben's savings over time.
| Days | 8 | 12 | 16 | |
|---|---|---|---|---|
| Amount saved | 7 | 14 | 21 |
Graph the amount each of them have saved over time.
Explain how to determine who is saving more per day.
Consider the proportional relationship shown between hours spent fishing and the number of fish caught displayed in the table below:
Sketch the graph of the proportional relationship.
Find the unit rate of the relationship.
By considering parts (a) and (b), make a conclusion about the number of fish caught.
| Hours | 4 | 8 | 12 |
|---|---|---|---|
| Fish caught | 9 | 18 | 27 |
The original of a printed image measures 8.5 \text{ cm} in width and 51 \text{ cm} in length. When a customer wants to print a copy of the original they are offered prints in various sizes, but the width and length are in the same ratio as the original so that the photo does not appear distorted.
Solve for the value of k, the constant of proportionality.
Find the length of a copy of the original if the width of the copy is 14 \text{ cm} .
The height, h, of a regular tetrahedron varies directly with its side length, s. A particular tetrahedron with a side length of 3 \text{ cm} has a height of 2.45.
Find the constant of proportionality, k, to two decimal places.
Express h in terms of s.
Find the height of a different tetrahedron with a side length of 2 \text{ cm}.