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

Consider the graph of a direct variation.

What is the relationship between the input

x

and output

y ?

A B C D The output is two more than the input . The output is four less than the input . The output is three times the input . The output is one third the input .

We will start by recalling the general form of a direct variation equation since we know that the graph represents a direct variation. y=kx Here, k is the constant of variation. To find its value, we can use any point on the given graph. By doing this, we will also determine the relationship between the input x and the output y. For simplicity, let's select the point (1,3).

Now, we will substitute x= 1 and y= 3 into the general formula to find the constant of variation.

The constant of variation is 3. Let's now write our direct variation equation. y=kx substitute y= 3x Now that we know the constant of variation, we can conclude that the output y is exactly the three times the input x. Choosing any other point would give us exactly the same result. Therefore, the answer is option C.

The following graph shows the average fuel consumption in liters per hour for four different brands of cars when the vehicles are driven at an average speed of $100$ kilometers per hour.

Which of the graphs represents the consumption of the vehicle that consumes

6

liters per hour?

A B C D K L M N

We know that one of the vehicles consumes 6 liters of fuel per hour when it is driven at an average speed of 100 kilometers per hour. This means we are looking for a direct variation graph that passes through the point (1, 6).

As we can see, the orange graph L passes through the point (1,6). This means that after 1 hour, the vehicle has consumed 6 liters of fuel. The answer is option B.

In the diagram, the graph of a direct variation and two points are shown.

graph of a direct variation and points (0,0) and (-1,3)

What is the constant of variation for the equation?

What is the slope of the line that passes through the given two points?

We will start by recalling the general form of a direct variation equation. y= kx In this form, k represents the constant of variation. With this in mind, let's have a look at the given direct variation equation. y= - 3x Since k= -3, the value of the constant of variation is -3.

To find the slope of the line that passes through the given points, we will substitute the points into the Slope Formula. m = y_2-y_1/x_2-x_1 In this case, the given points are (-1,3) and (0, 0). Note that when calculating slope, it does not matter which point we choose to use as (x_1,y_1) or (x_2,y_2).

The slope of the line that passes through the points is also - 3.

Suppose $y$ varies directly with $x$ and that $y = 4$ when $x = 14 .$

Write a direct variation equation that relates

x

and

y .

Find

y

when

x = - 21 .

Functions where y varies directly with x, known as direct variation equations, follow a specific format. y= mx In this form, m≠ 0. We can determine the constant of variation m by substituting the given values for x and y into the equation. Let's do it!

Now that we have the constant of variation, we can write our equation. y= 2/7x

Now that we have the direct variation equation, we can find any value of x or y when we are given the other value. In this case, we are looking for y when x=-21.

For the equation y= 27x, when x=-21, the value of y is -6.

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

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