Understanding Direct Variation Graphs and Proportional Variables

$x$	$y = 2 x + 1$
$1$	$3$
$2$	A
$3$	B
$4$	$9$
$5$	C

x

y = 2 x + 1

1

3

2

3

4

9

5

$x$	$y = x + 1$
$- 1$	$0$
$0$	$1$
$1$	$2$
$2$	$3$
$3$	$4$

x

y = x + 1

- 1

0

0

1

1

2

2

3

3

4

$g$	$0.5 g$	$p = 0.5 g$
$0$	$0.5 (0)$	$0$
$1$	$0.5 (1)$	$0.5$
$2$	$0.5 (2)$	$1$
$3$	$0.5 (3)$	$1.5$
$4$	$0.5 (4)$	$2$

g

0.5 g

p = 0.5 g

0

0.5 (0)

0

1

0.5 (1)

0.5

2

0.5 (2)

1

3

0.5 (3)

1.5

4

0.5 (4)

2

$x$	$\frac{1 0}{x}$	$y = \frac{1 0}{x}$
$1$	$\frac{1 0}{1}$	$10$
$2$	$\frac{1 0}{2}$	$5$
$3$	$\frac{1 0}{3}$	$\approx 3.3$
$4$	$\frac{1 0}{4}$	$2.5$
$5$	$\frac{1 0}{5}$	$2$
$6$	$\frac{1 0}{6}$	$\approx 1.7$
$7$	$\frac{1 0}{7}$	$\approx 1.4$
$8$	$\frac{1 0}{8}$	$1.25$
$9$	$\frac{1 0}{9}$	$\approx 1.1$
$10$	$\frac{1 0}{1 0}$	$1$

x

\frac{1 0}{x}

y = \frac{1 0}{x}

1

\frac{1 0}{1}

10

2

\frac{1 0}{2}

5

3

\frac{1 0}{3}

\approx 3.3

4

\frac{1 0}{4}

2.5

5

\frac{1 0}{5}

2

6

\frac{1 0}{6}

\approx 1.7

7

\frac{1 0}{7}

\approx 1.4

8

\frac{1 0}{8}

1.25

9

\frac{1 0}{9}

\approx 1.1

10

\frac{1 0}{1 0}

1

Determine if each equation represents a direct variation or not.

0.6 x - 2 y = 4

\frac{x}{4} + \frac{y}{8} = 0

If there is a direct variation between the variables x and y, then we can represent this relation with the following equation. y=kx In this form, k ≠ 0. We will rewrite the given equation to isolate y on the left hand side. Let's do it!

Notice that we cannot rewrite the equation in the form y=kx because we have a constant term -2 on the right-hand side. Therefore, the equation does not represent a direct variation.

We will apply the same process to isolate the variable y on the left-hand side of the equation. Let's do it!

Since we can write the equation in the form y=kx, where k=-2 in this case, the given equation does represent a direct variation.

A certain indoor playground is equipped with trampolines, slides, swings, and a ball pool. The price to play at the playground changes with the amount of time spent there. The table below shows how the price of play changes with respect to time.

Time (minutes)	$15$	$30$	$45$	$60$	$75$
Price (dollars)	$3$	$6$	$9$	$12$	$15$

Write an equation that relates the time

x

at the playground to the cost

y

of playing.

How long did a kid play at the playground if they spent

$ 20 ?

We are asked to write an equation relating the time x at the playground to the price y of playing there. To do so, we will first examine how these variables change accordingly. Time:& 15 +15 30 +15 45 ... Price:& 3 +3 6 +3 9 ... Notice that the price increases by 3 dollars for every 15 minutes of play. This means that there is a constant rate of change between the duration of play and the price. Therefore, we can express the relation between these variables with a direct variation. y= mx In this form, m ≠ 0 and is the constant of variation. To find its value, we can substitute one of the values from the table into the equation. Let's randomly choose the second time and price value, $6 for 30 minutes.

Now that we know the constant of variation, we can write a direct variation equation that relates the time x of play to its price y. y= 0.2x

To find the time the kid spent at the playground if the cost was $ 20, we can use the equation that we found in the previous part. y=0.2x Let's substitute y=20 into the direct variation equation and solve for x.

For $20, the kid played at the playground for 100 minutes.

A map of Italy is scaled so that a length of

4.5

inches represents

90

miles. How far apart are Bologna and Rome if they are

11.9

inches apart on the map?

The number of inches on the map between Bologna and Rome varies directly with the number of miles between the cities. We must write a direct variation equation, but we first need to determine the constant of variation. We know that that 4.5 inches represent 90 miles, so we can write the following ratio of miles to inches. 90 miles/4.5 inches We can simplify this value by dividing both numbers by 4.5. When we have 1 inch in the denominator, we can use it as a constant of variation. 90 miles/4.5/4.5 inches/4.5=20 miles/1 inch Now we can write the equation. Let's call the distance in miles between the cities M and the distance in inches i. M=20i To find how far apart the cities are in miles, we can substitute the given number of inches for i into the equation and simplify.

The cities are 238 miles apart.

Direct Variation and Proportional Relationships

Catch-Up and Review

Moving a Point Along a Line

Direct Variation

Graphing a Direct Variation

Hint

Solution

Writing a Direct Variation

Hint

Solution

Writing a Direct Variation Knowing a Point

Hint

Solution

Writing a Direct Variation Knowing Its Graph

Hint

Solution

Finding the Constant of Variation

Modeling With Direct Variation

Hint

Solution

Inverse Variation

Direct Variation and Proportional Relationships

Recommended exercises

	10 Theory slides
	8 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions