Math / Algebra

QuestionWhen proportional relationships are graphed, the points the line runs through can be used to find the constant of proportionality.
This line runs through points $(2,2),(4,4),(6,6)$ , and $(8,8)$ . First, find the proportion of this relationship by choosing one point and inserting its coordinates into the proportion equation. $k=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \text { or } k=\frac{4-2}{4-2}=\frac{2}{2}=1$
The constant of proportionality for this line is 1.
Find the constant of proportionality for each graph. I. $k=$ $\qquad$ 2. $k=$ $\qquad$ b $k=$ $\qquad$ $k=$ $\qquad$ Math

Studdy Solution

STEP 1

What is this asking? We need to find the constant of proportionality, often denoted as $k$ , for a bunch of graphs of proportional relationships.
It's like figuring out how steep each line is! Watch out! Make sure you pick points on the line, and don't mix up your $x$ and $y$ coordinates!

STEP 2

1. Understand Constant of Proportionality
2. Calculate $k$ for Graph I
3. Calculate $k$ for Graph a
4. Calculate $k$ for Graph b
5. Calculate $k$ for Graph 2

STEP 3

The constant of proportionality, $k$ , tells us how much the $y$ value changes for every change in the $x$ value.
It's like the line's "growth rate"!

STEP 4

We can find $k$ using any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line with the formula: $k = \frac{y_2 - y_1}{x_2 - x_1}.$

STEP 5

Graph I has the points ( $2, 1$ ) and ( $4, 2$ ).
Let's plug these into our formula.
We'll call $(2, 1)$ our first point $(x_1, y_1)$ and $(4, 2)$ our second point $(x_2, y_2)$ .

STEP 6

So, $x_1 = \mathbf{2}$ , $y_1 = \mathbf{1}$ , $x_2 = \mathbf{4}$ , and $y_2 = \mathbf{2}$ .
Plugging these values into our formula gives us: $k = \frac{\mathbf{2} - \mathbf{1}}{\mathbf{4} - \mathbf{2}} = \frac{1}{2}.$

STEP 7

Graph a has the points ( $2, 2$ ) and ( $4, 4$ ).
Let $(x_1, y_1) = (2, 2)$ and $(x_2, y_2) = (4, 4)$ .

STEP 8

Using the formula: $k = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1.$

STEP 9

Graph b has the points ( $2, 4$ ) and ( $4, 8$ ).
Let $(x_1, y_1) = (2, 4)$ and $(x_2, y_2) = (4, 8)$ .

STEP 10

Applying the formula: $k = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2.$

STEP 11

Graph 2 has the points ( $2, 5$ ) and ( $4, 10$ ).
Let $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (4, 10)$ .

STEP 12

Calculating $k$ : $k = \frac{10 - 5}{4 - 2} = \frac{5}{2}.$

STEP 13

I: $k = \frac{1}{2}$ a: $k = 1$ b: $k = 2$ 2: $k = \frac{5}{2}$

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Studdy Solution

STEP 1

What is this asking? We need to find the constant of proportionality, often denoted as $k$ , for a bunch of graphs of proportional relationships.
It's like figuring out how steep each line is! Watch out! Make sure you pick points on the line, and don't mix up your $x$ and $y$ coordinates!

STEP 2

1. Understand Constant of Proportionality
2. Calculate $k$ for Graph I
3. Calculate $k$ for Graph a
4. Calculate $k$ for Graph b
5. Calculate $k$ for Graph 2

STEP 3

The constant of proportionality, $k$ , tells us how much the $y$ value changes for every change in the $x$ value.
It's like the line's "growth rate"!

STEP 4

We can find $k$ using any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line with the formula: $k = \frac{y_2 - y_1}{x_2 - x_1}.$

STEP 5

Graph I has the points ( $2, 1$ ) and ( $4, 2$ ).
Let's plug these into our formula.
We'll call $(2, 1)$ our first point $(x_1, y_1)$ and $(4, 2)$ our second point $(x_2, y_2)$ .

STEP 6

So, $x_1 = \mathbf{2}$ , $y_1 = \mathbf{1}$ , $x_2 = \mathbf{4}$ , and $y_2 = \mathbf{2}$ .
Plugging these values into our formula gives us: $k = \frac{\mathbf{2} - \mathbf{1}}{\mathbf{4} - \mathbf{2}} = \frac{1}{2}.$

STEP 7

Graph a has the points ( $2, 2$ ) and ( $4, 4$ ).
Let $(x_1, y_1) = (2, 2)$ and $(x_2, y_2) = (4, 4)$ .

STEP 8

Using the formula: $k = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1.$

STEP 9

Graph b has the points ( $2, 4$ ) and ( $4, 8$ ).
Let $(x_1, y_1) = (2, 4)$ and $(x_2, y_2) = (4, 8)$ .

STEP 10

Applying the formula: $k = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2.$

STEP 11

Graph 2 has the points ( $2, 5$ ) and ( $4, 10$ ).
Let $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (4, 10)$ .

STEP 12

Calculating $k$ : $k = \frac{10 - 5}{4 - 2} = \frac{5}{2}.$

STEP 13

I: $k = \frac{1}{2}$ a: $k = 1$ b: $k = 2$ 2: $k = \frac{5}{2}$

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Studdy Solution

STEP 1

What is this asking? We need to find the *constant of proportionality*, often denoted as kkk, for a bunch of graphs of proportional relationships.It's like figuring out how steep each line is! Watch out! Make sure you pick points *on* the line, and don't mix up your xxx and yyy coordinates!

STEP 2

1. Understand Constant of Proportionality2. Calculate kkk for Graph I3. Calculate kkk for Graph a4. Calculate kkk for Graph b5. Calculate kkk for Graph 2

STEP 3

The *constant of proportionality*, kkk, tells us how much the yyy value changes for every change in the xxx value.It's like the line's "growth rate"!

STEP 4

We can find kkk using any two points (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​) on the line with the formula: k=y2−y1x2−x1.k = \frac{y_2 - y_1}{x_2 - x_1}.k=x2​−x1​y2​−y1​​.

STEP 5

Graph I has the points (2,12, 12,1) and (4,24, 24,2).Let's plug these into our formula.We'll call (2,1)(2, 1)(2,1) our first point (x1,y1)(x_1, y_1)(x1​,y1​) and (4,2)(4, 2)(4,2) our second point (x2,y2)(x_2, y_2)(x2​,y2​).

STEP 6

So, x1=2x_1 = \mathbf{2}x1​=2, y1=1y_1 = \mathbf{1}y1​=1, x2=4x_2 = \mathbf{4}x2​=4, and y2=2y_2 = \mathbf{2}y2​=2.Plugging these values into our formula gives us: k=2−14−2=12.k = \frac{\mathbf{2} - \mathbf{1}}{\mathbf{4} - \mathbf{2}} = \frac{1}{2}.k=4−22−1​=21​.

STEP 7

Graph *a* has the points (2,22, 22,2) and (4,44, 44,4).Let (x1,y1)=(2,2)(x_1, y_1) = (2, 2)(x1​,y1​)=(2,2) and (x2,y2)=(4,4)(x_2, y_2) = (4, 4)(x2​,y2​)=(4,4).

STEP 8

Using the formula: k=4−24−2=22=1.k = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1.k=4−24−2​=22​=1.

STEP 9

Graph *b* has the points (2,42, 42,4) and (4,84, 84,8).Let (x1,y1)=(2,4)(x_1, y_1) = (2, 4)(x1​,y1​)=(2,4) and (x2,y2)=(4,8)(x_2, y_2) = (4, 8)(x2​,y2​)=(4,8).

STEP 10

Applying the formula: k=8−44−2=42=2.k = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2.k=4−28−4​=24​=2.

STEP 11

Graph 2 has the points (2,52, 52,5) and (4,104, 104,10).Let (x1,y1)=(2,5)(x_1, y_1) = (2, 5)(x1​,y1​)=(2,5) and (x2,y2)=(4,10)(x_2, y_2) = (4, 10)(x2​,y2​)=(4,10).

STEP 12

Calculating kkk: k=10−54−2=52.k = \frac{10 - 5}{4 - 2} = \frac{5}{2}.k=4−210−5​=25​.

STEP 13

I: k=12k = \frac{1}{2}k=21​ a: k=1k = 1k=1 b: k=2k = 2k=2 2: k=52k = \frac{5}{2}k=25​

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to find the constant of proportionality, often denoted as $k$ , for a bunch of graphs of proportional relationships.
It's like figuring out how steep each line is! Watch out! Make sure you pick points on the line, and don't mix up your $x$ and $y$ coordinates!

1. Understand Constant of Proportionality
2. Calculate $k$ for Graph I
3. Calculate $k$ for Graph a
4. Calculate $k$ for Graph b
5. Calculate $k$ for Graph 2

The constant of proportionality, $k$ , tells us how much the $y$ value changes for every change in the $x$ value.
It's like the line's "growth rate"!

We can find $k$ using any two points $(x_1, y_1)$ and $(x_2, y_2)$ on the line with the formula: $k = \frac{y_2 - y_1}{x_2 - x_1}.$

Graph I has the points ( $2, 1$ ) and ( $4, 2$ ).
Let's plug these into our formula.
We'll call $(2, 1)$ our first point $(x_1, y_1)$ and $(4, 2)$ our second point $(x_2, y_2)$ .

So, $x_1 = \mathbf{2}$ , $y_1 = \mathbf{1}$ , $x_2 = \mathbf{4}$ , and $y_2 = \mathbf{2}$ .
Plugging these values into our formula gives us: $k = \frac{\mathbf{2} - \mathbf{1}}{\mathbf{4} - \mathbf{2}} = \frac{1}{2}.$

Graph a has the points ( $2, 2$ ) and ( $4, 4$ ).
Let $(x_1, y_1) = (2, 2)$ and $(x_2, y_2) = (4, 4)$ .

Using the formula: $k = \frac{4 - 2}{4 - 2} = \frac{2}{2} = 1.$

Graph b has the points ( $2, 4$ ) and ( $4, 8$ ).
Let $(x_1, y_1) = (2, 4)$ and $(x_2, y_2) = (4, 8)$ .

Applying the formula: $k = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2.$

Graph 2 has the points ( $2, 5$ ) and ( $4, 10$ ).
Let $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (4, 10)$ .

Calculating $k$ : $k = \frac{10 - 5}{4 - 2} = \frac{5}{2}.$

I: $k = \frac{1}{2}$ a: $k = 1$ b: $k = 2$ 2: $k = \frac{5}{2}$